%Representations of the Bondi$-$Metzner$-$Sachs group in three %space$-$time dimensions in the Hilbert %topology On the representation theory of the Bondi$-$Metzner$-$Sachs group and its variants in three space$-$time dimensions %\thanks{Grants or other notes %about the article that should go on the front page should be %placed here. General acknowledgments should be placed at the end of the article.
Evangelos Melas %
% Second Author %etc.
The original Bondi$-$Metzner$-$Sachs (BMS) group $B$ is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4$-$dim space$-$times. As such, $B$ is the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973, with this motivation, P. J. McCarthy classified all relativistic $B$$-$invariant systems in terms of strongly continuous irreducible unitary representations (IRS) of $B$. Here we introduce the analogue $B(2,1)$ of %Bondi$-$Metzner$-$Sachs group the BMS group $B$ in 3 space$-$time dimensions. $B(2,1)$ itself admits thirty$-$four analogues both real in all signatures and in complex space$-$times. In order to find the IRS of both $B(2,1)$ and its analogues we need to extend Wigner$-$Mackey's theory of induced representations. The necessary extension is described and is reduced to the solution of three problems. These problems are solved in the case where $B(2,1)$ and its analogues are equipped with the Hilbert topology. The extended theory is necessary %generalizations in order to construct the IRS of both $B$ and its analogues in any number $d$ of space$-$time dimensions, $d\geq3$, and also in order to construct the IRS of their supersymmetric counterparts. %In fact there are two variants of this group %which are isomorphic to each other. We use the extended theory to obtain the necessary data in order to construct the IRS of $B(2,1).$ %for both variants. The main results of the representation theory are: %Here, we construct the IRS of $B(2,1)$, the analogue of $B$, in 3 space$-$time dimensions. The IRS are induced from ``little groups'' which are compact. The finite ``little groups'' are cyclic groups of even order. %The %``little groups'' are the same for both variants but the %corresponding invariant ``supermomenta'' are different. The inducing construction is exhaustive notwithstanding the fact that $B(2,1)$ %neither of them is not locally compact in the employed Hilbert topology. \keywords{General Relativity \and Representation theory \and Infinite dimensional Lie groups \PACS{04 \and 03 \and 02 \and 11 \subclass{83 \and 81 \and 22E65 \and 22E70 %\and 20C35
1703.05980.tex
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\begin{document}

%\begin{center}
%  {



\title{
%Representations of the Bondi$-$Metzner$-$Sachs  group in three %space$-$time dimensions in the Hilbert
%topology
On the representation theory of the
Bondi$-$Metzner$-$Sachs  group and its variants
in three space$-$time dimensions
%\thanks{Grants or other notes
%about the article that should go on the front page should be
%placed here. General acknowledgments should be placed at the end of the article.}
}
%\subtitle{Do you have a subtitle?\\ If so, write it here}

%\titlerunning{Short form of title}        % if too long for running head
%}

%\end{center}





\author{Evangelos Melas
%\and
%        Second Author %etc.
}

%\authorrunning{Short form of author list} % if too long for running head

\institute{Evangelos Melas \at
              University of Athens \at
              Department of Economics \at
              Unit of Mathematics and Informatics \at
              Sofokleous 1 \at
              10559 Athens \at
              Greece \\
              Tel.: +030-210-3689403\\
              Fax: +030-210-3689471\\
              \email{emelas@econ.uoa.gr}           %  \\
%             \emph{Present address:} of F. Author  %  if needed
%           \and
%           S. Author \at
%              second address
%
}

\date{Received: date / Accepted: date}
% The correct dates will be entered by the editor


\maketitle

\begin{abstract}
The  original  Bondi$-$Metzner$-$Sachs (BMS) group $B$ is the common asymptotic symmetry group of all asymptotically flat Lorentzian radiating 4$-$dim space$-$times. As such, $B$ is  the best candidate for the universal symmetry group of General Relativity (G.R.). In 1973,  with this motivation, P. J. McCarthy classified all relativistic $B$$-$invariant systems in terms of strongly continuous irreducible unitary representations (IRS) of $B$. Here we introduce the analogue $B(2,1)$  of %Bondi$-$Metzner$-$Sachs group
the BMS group $B$
in 3 space$-$time dimensions. $B(2,1)$ itself
admits thirty$-$four  analogues both real
in all signatures and in complex space$-$times.
In order to find the IRS of both $B(2,1)$ and its analogues
we need to extend Wigner$-$Mackey's theory of induced
representations. The necessary extension is
described and is
reduced to the
solution of three problems. These  problems are solved in
the case where $B(2,1)$ and its analogues are equipped
with the Hilbert topology. The extended theory is necessary
%generalizations
in order to construct the IRS
of both $B$ and its analogues in any number $d$
of space$-$time dimensions, $d\geq3$, and also in order to construct the IRS
of their supersymmetric counterparts.
%In fact there are two variants of this group
%which are isomorphic to each other.
We use the extended theory  to obtain the necessary data in order to construct the IRS
of $B(2,1).$
%for both variants.
The main results of the representation theory are:
%Here, we construct the IRS of $B(2,1)$, the analogue of $B$, in 3 space$-$time dimensions.
The IRS are induced from ``little groups'' which are compact. The finite  ``little groups'' are cyclic groups of even order.
%The
%``little groups'' are the same for both variants but the %corresponding invariant ``supermomenta'' are different.
The inducing construction is exhaustive notwithstanding the fact that $B(2,1)$
%neither of them
is not locally compact in the employed Hilbert topology.








\keywords{General Relativity \and Representation theory \and Infinite dimensional Lie groups}
\PACS{04 \and  03 \and 02 \and 11}
\subclass{83 \and 81 \and 22E65 \and 22E70
%\and 20C35
}
\end{abstract}

\section{Introduction}
\label{intro}
The
%Bondi$-$Metzner$-$Sachs
BMS group $B$ is the common asymptotic group of all
curved real Lorentzian radiating space$-$times which are  asymptotically flat in future null directions \cite{Bondi,Sachs1}, and is the best candidate for the universal symmetry group of G.R..

The BMS group $B$ of general relativity was first
discovered \cite{Bondi,Sachs1} not as a transformation group of
(exact) global diffeomorphisms of a fixed manifold,
but as a pseudo$-$group of local diffeomorphisms
(``asymptotic isometries") of the asymptotic region
of Lorentzian radiating space$-$times which are asymptotically
flat in lightlike future directions.

However, Penrose \cite{Pen1,Pen2,Pen3} showed that, by ``going to infinity", $B$ could be interpreted as an (exact) global transformation group $B \times \Im^{+} \longrightarrow \Im^{+}$ of the ``future null boundary" $\Im^{+}$, or, of the
so called ``future null infinity", of the space$-$times concerned. Furthermore, he gave \cite{Pen1,Pen2}  a geometric structure to $\Im^{+}$, the ``strong conformal geometry" for which the
transformation group is precisely the group of automorphisms
\cite{Pen3}.









In 1939 Wigner laid the foundations
of special relativistic quantum mechanics \cite{Wigner} and relativistic quantum field theory by constructing
the Hilbert space
%strongly continuous unitary irreducible representations (IRS)
IRS of the (universal cover) of the Poincare group $P$.

%In a quantum setting
The universal property of $B$ for G.R.
%general relativity
makes it reasonable to attempt
to lay a similarly firm foundation for quantum gravity by following through the analogue of
Wigner's programme with $B$ replacing $P$. Some years ago McCarthy constructed explicitly \cite{mac3,mac8,mac4,mac5,mac6,mac7,Crampin2,Mac}   the IRS of $B$ for exactly this purpose.
This work was based on G.W.Mackey's pioneering work on group representations \cite{Mackey0,Mackey00,Mackey,Mackey1,Simms,Isham}; in particular
it was based on an extension \cite{mac6} of G.W.Mackey's  work to the relevant
infinite dimensional case.

%McCarthy's work extended G.W.Mackey's  work to the relevant
%infinite dimensional case.

It is difficult to overemphasize the importance of Piard's results \cite{Piard1,Piard2} who soon afterwards proved that \it{all} \normalfont the IRS of $B$,
when this is equipped with the Hilbert topology, are derivable by the inducing construction.
This proves the exhaustivity of McCarthy's list of representations and renders his results even more important.









However, in quantum gravity, complexified or euclidean versions of G.R. are frequently considered and
the question arises: Are there similar symmetry groups for these versions of the theory? McCarthy constructed \cite{mac1}, in abstract form,
all possible analogues of $B$, both real and in any signature, or complex, with all possible notions of asymptotic flatness ``near infinity''. There are, in fact, forty$-$one such groups.
One of them is $B(2,2)$, the asymptotic symmetry group of all curved real ultrayperbolic space$-$times which are asymptotically flat in null directions.
%Let us denote these groups collectively, $B$ included, by   %$\mathcal G$.


These abstract constructions were given in a quantum setting; the paper \cite{mac1} was concerned with finding the IRS of these groups
%$\mathcal G$
in Hilbert spaces (especially for the complexification $ \mathcal C \mathcal B $ of $B$ itself). It has been argued \cite{mac1,Mel1}
%was argued
that these Hilbert space representations are
%were
related to elementary particles and quantum gravity (via gravitational instantons). The IRS of $B(2,2)$ were constructed in \cite{macMel,Mel5}.
HB, a subgroup of $B(2,2)$ which arises naturally in the construction of the analogues of  $B$ by McCarthy \cite{mac1}, and which, surprisingly, remained unnoticed by him, was introduced in \cite{Melas1}. The representation theory of HB was commenced in \cite{Mel6,Mel7}.







Here,
we initiate                                      this programme for 3$-$dim G.R..
We define the analogue $B(2,1)$ of the Bondi$-$Metzner$-$Sachs (BMS) group
in 3 space$-$time dimensions.
%In fact there are  two variants \cite{Melas},
%$B(2,1)$ and $\mathcal B(2,1)$,  of this group:
%In $B(2,1)$  the ``supertranslations'' are defined as even functions %on the
%circle $S^{1}$, whereas, in  $\mathcal B(2,1)$
%the ``supertranslations'' are defined as free functions on %$P_{1}(R)$,
%where $P_{1}(R) \equiv S^{1}/Z_{2}$ is the one$-$dimensional real %projective space
%(the circle quotient the antipodal map). $B(2,1)$ and $\mathcal %B(2,1)$ are
%isomorphic to each other.
%%We define $B(2,1)$, the analogue of $B$ in three
%%space$-$time dimensions.
We construct the IRS of
%both variants
of $B(2,1)$
%by endowing it
%them
%with
in the Hilbert topology.
McCarthy and Crampin argued convincingly
in \cite{Crampin2} that
in the case of the original
BMS group $B$, in 4
space$-$time dimensions,
IRS
%of $B$
in the Hilbert topology describe
%gravitating systems with
\it bounded \normalfont sources \cite{Crampin2}, i.e., $B-$elemenatry particles.
Moreover they pointed out that passing from
the coarser Hilbert topology to the finer
nuclear topology the effect is to obtain more
IRS which now also accommodate scattering states of $B-$elemenatry particles.


Their arguments apply also in the case of
$B(2,1)$, in 3
space$-$time dimensions, so we expect IRS of $B(2,1)$
in the Hilbert topology to describe
bounded sources, i.e., $B(2,1)-$elemenatry particles, whereas IRS in the
nuclear topology,
to describe also scattering states of $B(2,1)-$elemenatry particles. Besides
the Hilbert and the nuclear topologies there
is a  wide range
%\cite{Crampin2}
of
``reasonable'' topologies by which we can endow $B(2,1)$ with. These topologies are elaborated in Section \ref{top}.
$B(2,1)$ admits generalizations both real
in all signatures and in complex space$-$times
with all possible notions of asymptotic
flatness ``near infinity''. There are, in fact, thirty$-$five such groups including $B(2,1)$ \cite{Mel8}.


These thirty$-$five  groups have the general structure \cite{Mel8}
\be
\mathcal B =  \mathcal S(A, K) \rtimes_{T} G,
\ee
where $A$ is a locally compact space, $K$ is the field of real or complex numbers,
and $G$ is the group of global, conformal, orientation$-$preserving transformations
of the space  $A$   onto itself. $\mathcal S(A, K)$  is the space of
supertranslations, i.e., of generalized position$-$dependent translations.
The supertranslations $\mathcal S(A, K)$
 are $K-$valued functions defined on $A$, and they constitute a normal abelian
 subgroup of $\mathcal B.$  Moreover, $\mathcal S(A, K)$ has a vector space structure, over the
 field $K$, under point$-$wise addition and scalar multiplication.



 In the case of $B(2,1)$ $A$ is the circle $S^{1}$, $K$ is the field of real numbers, $G$ is the group
%  ${\rm S}{\rm L}(2,R)$
 $ S L(2,R)$
  of all real $2\times 2$ matrices with determinant one,
  and $\mathcal S(A, K)$ is  the space  of
%  ``free''
  real$-$valued
functions defined on $P_{1}(R)\simeq S^{1}$.
$P_{1}(R) $ is the one$-$dimensional real projective  space (the circle quotient the antipodal map).
Alternatively, as it is explained in Section \ref{sec1},
in a different realization of $ B(2,1)$,
 $S(A, K)$ is  the space  of  ``even'' real$-$valued defined on $ S^{1}$.






Ashtekhar $\&$ Hansen, in a unified treatment of asymptotic flatness in both null and spacelike directions for real lorentzian 4$-$dim space$-$times \cite{ass},
%examining both spacelike and null infinity for real lorentzian 4$-$dim space$-$times,
derived a BMS$-$like group, the \it Spi \normalfont group, which acts on the spacelike analogue of ``null infinity" $\Im^{+}$, the
``spacelike infinity". ``Spacelike infinity" is a 4$-$dim manifold, as opposed to ``null infinity" $\Im^{+}$ which is a 3$-$dim manifold.
One of the  thirty$-$five generalizations  of $B(2,1)$, is the group $B_{S}(2,1)= \mathcal S(S^{1} \times I, R) \rtimes_{T} SL(2,R)$,
which is the analogue of the  \it Spi \normalfont group  in
3$-$dim space$-$times. $I$ is an open interval of $R,$ and $\mathcal S(S^{1} \times I, R)$ is
the space of real$-$valued
functions defined on the product $S^{1} \times I$ which is homeomorphic to a cylinder in $R^{3}$.
In the case of 3$-$dim space$-$times ``spacelike infinity" is a 3$-$dim manifold, whereas ``null infinity" is a 2$-$dim manifold.





By following Penrose \cite{Pen1,Pen2,Pen3} we can interpret $B(2,1)$ as a transformation group $B(2,1) \times \Im^{+}$
of the ``future null infinity" of the Lorentzian 3$-$dim space$-$times involved.
Furthermore, we can  give   a geometric structure to $\Im^{+}$, the ``strong conformal geometry" for which the
transformation group is precisely the group of automorphisms. %True,
The assignment of the ``strong conformal geometry" to the
2$-$dim
$\Im^{+}$, in the case of 3$-$dim space$-$times,
follows Penrose's original  beautiful construction \cite{Pen1,Pen2,Pen3}
of the ``strong conformal geometry" in the case of 4$-$dim space$-$times.



However, the definition of the ``strong conformal geometry",  in the case of 3$-$dim space$-$times, does not come as an immediate corollary of Penrose's original construction
 of the ``strong conformal geometry",
in the case of 4$-$dim space$-$times, simply by
following through Penrose's construction and by
reducing on the way the number of spatial dimensions by one.
There are some subtleties involved when we pass from
4 space$-$time dimensions to 3 space$-$time dimensions.
In particular some care is needed,  in 3 space$-$time dimensions, in the appropriate definition
of the notion of the ``null angle",
introduced by Penrose \cite{Pen2,Pen3}, in order to establish
the ``strong conformal geometry".
% which is  indispensable for the construction of the ``strong %conformal geometry".
Here we do not elaborate further on this issue but we rather
restrict our attention  to the representation theory of $B(2,1)$ and its subtle underpinnings.





%The key new feature \cite{mac1} of the %original BMS group $B$,
%as well as of all possible analogues of $B$, %both real and
%in any signature, or complex, with all %possible notions
%of asymptotic flatness ``near infinity'', is %the enlargement
%of the finite dimensional vector space of %``translations''
%to the infinite dimensional

%$B(2,1)$ admits generalizations both real
%in all signatures and in complex space$-$times
%with all possible notions of asymptotic
%flatness ``near infinity''.



%These comments are contained implicitly in and %apply not only to B but also to b(2,1) and %their extensions and generalizations.


%The comments made in Sections \ref{sec2} and \ref{sec3} on %Wigner$-$Mackey's  representation theory we use in order to %find the IRS of $B(2,1)$ are not
%applicable only to $B(2,1)$ but they are also relevant and
%applicable to the thirty$-$four generalizations \cite{Mel8} %of $B(2,1)$ both real in all signatures and in complex %space$-$times.
%In fact the relevance and applicability of the
%comments made in Sections  \ref{sec2} and \ref{sec3}, and in %particular the extension of Wigner$-$Mackey's   theory we %introduce in Section \ref{sec3},
%permeates also the representation theory of the original
%BMS group $B$, as well as the representation theory of its %forty$-$one generalizations \cite{mac1}
%both real in all signatures and in complex space$-$times.
%%Last but not least
%In fact
%they
%%the comments made in Sections \ref{sec2} and \ref{sec3}
%are also applicable both to the generalizations of $B$ in any %number $d$
%of space$-$time dimensions, where $d\geq3$, and
%to their supersymmetric counterparts \cite{shaw}.
%% in any
%%in any number $d$
%%of space$-$time dimensions, where $d\geq3$.





%Moreover they pointed out that passing from
%a coarser (Hilbert) topology to a finer
%(nuclear) topology the effect is










%The paper closes with the explicit construction of the IRS of both %variants



%representation theory.

As it is explained in Section \ref{sec3} in
order to find the IRS  of any BMS group $B$,  as well as the IRS of any of its analogues,
 in any number $d\geq3$
of space$-$time dimensions, we cannot
apply Wigner$-$Mackey's theory as it is.
%In our approach in order to apply Wigner$-$Mackey's theory
We need to extend it.
The required extension, is reduced  to the solution of three problems, described in detail in Subsection
\ref{extension}. These three problems are named the First, the Second, and the Third problem, and they are
correspondingly, the ``little groups'' problem, the irreducibility problem, and the exhaustivity problem.


In Subsection
%\ref{orbits} and
\ref{little}
we find a solution to the First problem. The solution
applies to any topology $ B$  and its analogues come equipped with, in any number $d\geq3$
of space$-$time dimensions. The solution to the Second problem, given in
Subsection \ref{solutions}, applies to the Hilbert
and to the nuclear topologies. Finally, the solution to the Third problem, given in
Subsection \ref{solutions}, applies to the Hilbert
 topology only.
Therefore solution to all three problems
 posed by  the
required extension of Wigner$-$Mackey's theory are
given in the Hilbert topology, and consequently, a complete list of IRS of $B(2,1)$
can be constructed in this topology.



%The comments made in Sections \ref{sec2} and \ref{sec3} on %Wigner$-$Mackey's  representation theory we use


The extension of Wigner$-$Mackey's   theory we introduce in Section \ref{sec3}
in order to find the IRS of $B(2,1)$
is not
applicable only to $B(2,1)$
but it is also
%they are also
relevant and
applicable to the thirty$-$four generalizations \cite{Mel8} of $B(2,1)$, both real in all signatures and in complex space$-$times.
In fact the relevance and applicability of the
extended theory
%comments
%made in Sections  \ref{sec2} and \ref{sec3}, and in %particular the extension of Wigner$-$Mackey's   theory we %introduce in Section \ref{sec3},
permeates also the representation theory of the original
BMS group $B$, as well as the representation theory of its forty$-$one generalizations \cite{mac1},
both real in all signatures and in complex space$-$times.
%Last but not least
%In fact
%they
%%the comments made in Sections \ref{sec2} and \ref{sec3}
%are also applicable both to the generalizations of $B$ in any %number $d$
%of space$-$time dimensions, where $d\geq3$, and
%to their supersymmetric counterparts \cite{shaw}.




In fact by using the solutions to the
three problems in the Hilbert topology,
and thus the extended
Wigner$-$Mackey's   theory given in Section \ref{sec3},
% given in this paper
 we can construct, in the Hilbert topology,
% in principle
 all
the IRS of any BMS group $B$,  as well as the IRS of any of its (supersymmetric \cite{shaw}) analogues,
 in any number $d\geq3$
of space$-$time dimensions.
%The construction gives \it all \normalfont
%the IRS of $B$ when $B$ is equipped with the
%Hilbert topology.


In the other
topologies for $B$, described in
Section \ref{top}, the
solution to the
``little groups'' problem
given in Subsection \ref{little},
allows us to
pursue Wigner-Mackey's
inducing construction and obtain
a list of representations for
$B.$ We do not know if the
representations so obtained
are irreducible, and moreover, we do not know
 if  the inducing construction gives
all the representations of $B.$


In order to find if
%they
the representations  are
irreducible we need to solve in each case
(apart from the Hilbert topology case and the nuclear topology case) the irreducibility problem which is
stated in Subsection \ref{extension} and solved in Subsection
\ref{solutions} for the Hilbert and nuclear topologies.
Furthermore, in order to find if
we obtain all the representations  of $B$
by using Wigner$-$Mackey's inducing construction
we need to solve in each case
(apart from the Hilbert topology case)
the exhaustivity problem which is
stated in Subsection \ref{extension} and solved in Subsection
\ref{solutions} for the Hilbert topology.

%we need to solve in each case
%construction
%gives representations of $B$
%for which we do not know



% whereas in the other
%topologies for $B$ given in Section \ref{top},







%A complete list of IRS is obtained when
%$B$ is equipped with the Hilbert topology,
%whereas in other topologies,












To the best of our knowledge,
the  group theoretical approach  to quantum gravity advocated here, is
the only approach, in which a complete list of IRS of the appropriate
symmetry group, i.e., of $B$ and of
 any of its analogues,
 in any number $d\geq3$
of space$-$time dimensions,
% in our approach,
% in our approach or of $\mathcal C$
%in Isham's approach,
%can be
is constructed in a given (i.e. in the Hilbert)
topology.


The other approaches we are aware of
\cite{Isham,IshKak0,IshKak,Bar3,Bar4} are afflicted
by the inapplicability of the
Wigner$-$Mackey's inducing construction
which manifests itself in
the appearance of the ``little groups'' problem, the irreducibility problem, and the exhaustivity problem,
which persistently occur, in this or the other form,
%show up
in any group theoretical approach to
quantum gravity.



The main results of the representation theory  are:
% and
%construct in the Hilbert topology
%%for this case, the strongly continuous unitary irreducible %representations
%the IRS of $B(2,1)$, the analogue of $B$ in three space$-$time %dimensions.
%%, are
%%analysed in the Hilbert topology.
%It is proved that
%All
The
%induced
IRS of
%both variants
$B(2,1)$
are induced
%arise
from IRS of \it compact \normalfont ``little groups''.
The ``little groups''
%, are the same for both variants, and
are of two types:
1. Infinite connected Lie groups. 2. Non$-$connected finite discrete groups.
%However, the corresponding invariant  ``supermomenta'' are different %for the
%two variants.
The  non$-$connected finite discrete ``little groups'' are
cyclic groups of even order, which are symmetry groups of  regular
polygons in ordinary euclidean 2$-$space.
%%It follows that
%Some IRS of $B(2,1)$ are controlled by
%%the
%IRS of the
%finite symmetry groups, of even order, of regular polygons in ordinary euclidean 2$-$space.
%It is  proved that
The inducing construction is exhaustive
%for both variants
notwithstanding the fact that $B(2,1)$
%neither of them
is
\it not \normalfont
locally compact in the employed Hilbert topology.
%\it All \normalfont IRS of $B(2,1)$ are induced by
%the IRS of its ''little groups''
%%from
%%obtained by
%%Mackey's semidirect product theory
%notwithstanding the fact that $B(2,1)$
%is \it not \normalfont locally compact in the employed Hilbert topology.

%We give all the necessary information
%in order to construct the operators of the induced representations of
%$B(2,1)$.

























A terminological remark is here in order. Whenever, hereafter, mention is made of a representation of a group, it  will
always be taken to mean  \it strongly continuous \normalfont representation. If $\mathcal G$ is a topological group we say
that a representation $\mathcal U$ of $\mathcal G$
on a Hilbert space $\mathcal H$
is strongly continuous if $g_{n} \rightarrow g$, $n=1,2,3,... $,  in $\mathcal G$, implies $\| \mathcal U(g_{n})\psi \rightarrow \mathcal U(g)\psi   \|   $   for all $\psi \in \mathcal H.$



















This paper is organized as follows.
In Section 2 relation to other approaches is
discussed.
In  Section 3 $B(2,1)$ is introduced.
%both variants are introduced.
In Section 4 the bare essentials of  Wigner$-$Mackey's theory are given. In Section 5 an extension of Wigner$-$Mackey's theory, necessary to construct the IRS of $B(2,1),$
%both invariants,
%and in fact the IRS of any BMS group
and in fact the IRS of any BMS group $B$,  as well as the IRS of any of its (supersymmetric) analogues,
 in any number $d\geq3$
of space$-$time dimensions,
is developed.  In Section 6 a heuristic proof of the compactness of the ``little groups'' is given, and
 it is pointed out that besides the Hilbert topology, considered in this paper, there are other
``reasonable'' choices for the topology
of the supertranslation space. In Section 7
the necessary data in order to construct the operators
of the IRS of $B(2,1)$ are given.
%the results of the representation theory of both variants are %stated.

\section{ Relation to other approaches }
\label{other}

\noindent
The BMS group $B$
%and its variants,
%as well as their supersymmetric counterparts,
can be derived, in general \cite{Hol,Wald,Tan1,Tan2,Tan3}, in any number $d \geq 3$ of space$-$time dimensions,
by using the four methods \footnote{There is also Bramson's derivation
\cite{Bram} who defines the BMS group $B$ to be the group of
mappings of ``cone space'', introduced by him,
 onto itself which acts linearly on the space
of position vectors and which preserves the norm of each position vector.
Bramson's definition of $B$ is illuminating and interesting but is not relevant to this study and so we do not include it in the methods of derivation of $B.$ } specified below.
%in the three ways specified below.
The order of exposition reflects the chronological order in which they appeared
\cite{Bondi,Pen1,mac8,Ger}.

\begin{enumerate}
\item{As a pseudo$-$group of local diffeomorphisms
%(``asymptotic isometries")
%of the asymptotic region
%of Lorentzian space$-$times
which preserve the asymptotic form of the
Bondi$-$Van Der Berg$-$Metzner
class of
metrics \cite{Bondi}
%asymptotic region
of Lorentzian radiating space$-$times which are asymptotically
flat in lightlike future directions.}
\item{As
  an exact global transformation group $B \times \Im^{+} \longrightarrow \Im^{+}$ of the ``future null boundary" $\Im^{+}$ of the
Bondi$-$Van Der Berg$-$Metzner space$-$times. This group is defined to be
%an exact global transformation group of diffeomorphisms
%of a fixed manifold
%which is
%which is
the group of automorphisms of the ``strong conformal geometry",
 a geometric structure  given by Penrose \cite{Pen1}, to the ``future null boundary" $\Im^{+}$ of the space$-$times concerned.
%which is the group of
}

\item{As the group which results from the
%``Poincare group''
Poincare group
when the translations,
which can be put into bijective correspondence with a finite parameter family of homogeneous  functions of degree one,
%when the finite parameter family is
are replaced by an appropriate infinite parameter family of homogeneous functions of degree one \cite{mac8}.}








\item{As a group of diffeomorphisms of the ``future null boundary" $\Im^{+}$ of the
Bondi$-$Van Der Berg$-$Metzner space$-$times which leaves invariant the
``universal structure'' \cite{Ger}. The ``universal structure'' is
equivalent to the ``strong conformal geometry" introduced by Penrose.
The key point in this method is that the invariance
of the  ``universal structure'' under the
action of $B$ on $\Im^{+}$ is expressed not in terms
of the elements of the group $B$ itself, but instead,
\it
%expressed
in terms
of the linearization of the group $B$ close to its
identity element, i.e., in terms of the generators
of the Lie algebra of the group $B$.
}











\end{enumerate}


The three first methods of derivation
%when applied
give the group $B$, whereas the fourth method
%when applied
gives the Lie algebra of $B$. A word of caution is here in order.
Both $B$ and its (supersymmetric) analogues in any number $d\geq3$ of space$-$time dimensions are infinite dimensional
Lie groups. For  infinite dimensional
Lie groups, the exponential map, when it exists, from  a
neighbourhood of 0 in the Lie algebra of $B$ to a
neighbourhood of $e$ in $B$, is \it not \normalfont surjective.
Thus exponentiation of the Lie algebra of $B$ does not give, in general,
the connected component of the identity of $B$.
Therefore any conclusions drawn for $B$ by studying its Lie algebra
should be drawn
%taken on board
with this word of caution in mind.













One can also derive the variants of $B$,
%and its variants,
as well as their supersymmetric counterparts,
in any number $d\geq3$ of space$-$time dimensions,
%by using
with the use of the four methods stated above, by appropriately
modifying in each case, the class of metrics in the first
method, the notion of ``strong conformal geometry" in the second method,
the appropriate infinite parameter family of homogeneous functions of degree one in the
third method, and finally, the notion of ``universal structure"
in the fourth method.
%The quotes in ``Poincare group'' in the third way indicate that
%this term  refers both to real space$-$times in any signature
%and in complex space$-$times in any number of $d\geq 3$  space$-$time %dimensions.
\noindent
%The first way is based on considering the asymptotic form of the
%Bondi$-$Van Der Berg$-$Metzner
%class of
%metrics
%%metric
%of radiating
%space$-$times and so we call it the asymptotic way.
%The second way is geometric in character
%and so we call it the geometric way.
%The inception of the third way is
%algebraic in character and so we call it the algebraic way.








Each method
%way
has its own merits and drawbacks. For example,
it appears \cite{Wald}, that
%Therefore,
in odd
space$-$time dimensions, the second and the fourth method of derivation
%conformal method of derivation
are not applicable in the case of \it radiative \normalfont space$-$times.
This restriction is not applicable to the case of three space$-$time
dimensions we are studying in this paper,
%here
because in three space$-$time
dimensions
%this case
%the vacuum
%solutions of Einstein's equations are necessarily flat, and
there are no propagating degrees of freedom, i.e., there is no gravitational
radiation.
% it does not appear that conformal null infinity can be in any
%way useful for describing radiation.




%We note in passing that
%in any case one should  proceed with care when he wants to
%derive the BMS group $B$, or any of its (supersymmetric) %analogues, in any number
%$d\geq3$ of space$-$time dimensions, by using the four %methods of derivation.



%As we pointed out a method  may not applicable in a %particular case, and moreover, it may be the case that when %it is applicable gives rise to a BMS group which does not %have the physical significance of the original BMS group %$B$, namely, it is
%not the asymptotic group of isometries of a class of metrics %of
%asymptotically flat \it radiating \normalfont space$-$times.
%This happens for instance in the case of the analogues of %the
%$B$ in euclidean and
%%the applicability of the four ways of derivations deserves %further study.


%is far from being solved
%and it is a problem on its own right.ts



Here we introduce
%, in the algebraic way,
the analogue $B(2,1)$
%$B(2,1)$
of %Bondi$-$Metzner$-$Sachs group
the BMS group $B$
in 3 space$-$time dimensions
%in the algebraic way
by using
for the first time the algebraic way of derivation.
To the best of our knowledge,
the asymptotic symmetry group in 3 space$-$time
dimensions
%the analogue   of
%%$B(2,1)$
%of %Bondi$-$Metzner$-$Sachs group
%the BMS group $B$
%in 3 space$-$time dimensions
was firstly considered
in \cite{Bij}, a work which is entirely classical in aims and scope.
In particular  no IRS were considered in this study.


%However, in this investigation, which falls into the realm of the
%geometric  way of derivation, the authors concentrated entirely on the Lie %algebra of
% the analogue   of
%%$B(2,1)$
%of %Bondi$-$Metzner$-$Sachs group
%the BMS group $B$
%in 3 space$-$time dimensions. They did not write explicitly the group
%and they did not specify its semidirect product structure.

%The authors, in this investigation, constructed
%%concentrated entirely on the Lie algebra of
% the analogue   of
%%$B(2,1)$
%of %Bondi$-$Metzner$-$Sachs group
%the BMS group $B$
%in 3 space$-$time dimensions by using the second method
%of derivation.


In this investigation the authors studied
gravitational waves with a space-translation Killing field. In this case, they noticed, the 4$-$dimensional Einstein vacuum equations are
equivalent to the 3$-$dimensional Einstein equations with certain matter sources.
%A key point
%The authors used the fourth method of derivation.
The key point
in their study
which determined the structure of the asymptotic
symmetry group they derived,
%and was a forerunner for future developments,
%undertaken by Barnich and coworkers (see e.g.
%\cite{Bar3,Bar4,Bar1,Bar2} and references therein),
is the following. The vacuum gravitational wave solutions
they considered in 4 space$-$time dimensions are not asymptotically flat either at spatial
or null infinity. This led to weaker fall off conditions
for the matter sources in the equivalent system of
Einstein equations with matter sources.



The weaker fall off conditions led in turn to the enlargement of the asymptotic
symmetry group they derived.
%The authors
%in order to
%derived the asymptotic symmetry group with the fourth method
%of derivation.
%The authors  derived the analogue of the BMS group $B$
%in 3 space$-$time dimensions as
%the group of automorphisms of the
%``universal structure'' of ``null infinity" $\Im^{+}$,
%the ``universal structure''  being Penrose's
%``strong conformal geometry". The authors
%expressed the conditions for the invariance
%of the  ``universal structure'' under the
%action of the group on $\Im^{+}$, not in terms
%of the elements of the group itself, but instead,
%\it
%expressed
%in terms
%of the linearization of the group close to its
%identity element, i.e., in terms of the generators
%of the Lie algebra of the group. \normalfont
It led in particular to
%had as a consequence
the enlargement of $SL(2,R)$
to the full group $Diff(S^{1})$ of the orientation preserving diffeomorphisms of the
circle.
% As we explain in Subsection \label{sec2.3}
% the group
 $SL(2,R)$ acts on $P_{1}(R)\simeq S^{1} $, the one-dimensional real projective  space (the circle quotient the antipodal map), this
 is going to be of interest in  Subsection \ref{sec2.3}.
As a result of this, the group obtained by the authors, by using the fourth method of derivation, is not
the semidirect product of supertranslations, i.e., of the additive group of functions on  the
circle, with $SL(2,R)$, which is  the group $B(2,1)$ considered in this
paper, but it is instead the semidirect product of the supertranslations
%additive group of functions on  the circle 
with $Diff(S^{1})$.



The analogue of the BMS group $B$
in 3 space$-$time dimensions with the first method of derivation was given in \cite{Bar2} and \cite{Bar3}.
In \cite{Bar1} the authors 
advocated the view put forward e.g. in
%motivated by the results  of  
\cite{Str,Str1,Graham,Aha,Boer,Sol}, namely that 
%in two$-$dimensional conformal field theories 
physics should be constrained by infinitesimal 
symmetry transformations that are not necessarily 
globally well defined. 


With this motivation
 the authors initiated a research program
in \cite{Bar1} whose objective is to materialize this
thesis in Einstein's G.R. in asymptotically flat space$-$times.
Firstly they considered the 4$-$dim case \cite{Bar1,Bar2}, and then 
they examined the  3$-$dim case \cite{Bar2,Bar3,Bar4}.
% and then  
%motivated by the results  of  \cite{Str,Str1} and following  the %reasoning of \cite{Graham,Aha,Boer,Sol},
%initiate 


%With this motivation they initiated a research program
%in \cite{Bar1} where they take 
%%where they took 
%the symmetry  algebra at null infinity to be not the BMS Lie %algebra, i.e.,
%% $\bf bms_{4}$,
%% i.e.,
%  the
%semi$-$direct sum  of the Lorentz algebra
%%$\bf so(3,1)$ 
%with the infinitesimal supertranslations, but they take it instead %to be the semi$-$direct sum  of the infinitesimal
%\it local \normalfont conformal transformations  of  the 2$-$sphere %with the abelian ideal of supertranslations, and now both
%factors are infinite$-$dimensional.


%In the same spirit 
In particular, in three space$-$time dimensions, the authors 
%in \cite{Bar3,Bar4}
studied \cite{Bar3,Bar4}, by using Wigner$-$Mackey's theory,
%in three space$-$time dimensions, 
the projective representations  of BMS$_{3}$, the semidirect product of
$Diff(S^{1})$  with the supertranlations, the additive group 
of functions on the circle. 
%This is the group denoted by 
%BMS$_{3}$ by the authors. 
%Therefore
Thus both factors of  BMS$_{3}$ are infinite dimensional. 
There is no overlap 
between this paper and \cite{Bar3,Bar4} because the 
group $B(2,1)$ we are considering in this paper is different from the group BMS$_{3}$ whose representation theory is undertaken in \cite{Bar3,Bar4}.

The representation theory of  BMS$_{3}$ is also afflicted 
by the inapplicability of the
Wigner$-$Mackey's inducing construction
%which manifests itself in
%the appearance of the ``little groups'' problem, the irreducibility %problem, and the exhaustivity problem,
which persistently occurs, as we pointed out in Section 1, in this or the other form,
%show up
in any group theoretical approach to
quantum gravity. 


In the case of BMS$_{3}$ the  
inapplicability of the
Wigner$-$Mackey's inducing construction
 manifests itself in
the appearance of the exhaustivity problem 
and in the construction of quasi$-$invariant$-$measures
in the appropriate orbits \cite{Bar4}. No solution is 
given by the authors to these problems and are left for
future  consideration \cite{Bar4}.   
In the case of $B(2,1)$ solution to the exhaustivity problem
in the Hilbert topology is given in Subsection \ref{solutions}
and quasi$-$invariant$-$measures are constructed in Subsection
\ref{invmeas}.
 

A last remark is here in order regarding 
$B(2,1)$ and BMS$_{3}$. $B(2,1)$ is a proper subgroup of BMS$_{3}$. %This fact though 
However, this does not render the study of the representations
of $B(2,1)$ superfluous. In general, the following holds: Let $K$ be a group and $H$ be a subgroup of $K$. Let $T(k)$ be
an irreducible representation of the group $K$. Let $T_{H}(k)$ be the restriction of $T(k)$ on the subgroup
$H$. In general, the representation $T_{H}(k)$ is not irreducible. Moreover, there are irreducibles of $H$ which
cannot be extended to the whole group $K$ .














As we pointed out, nuclear topology IRS of $B$
appear to allow scattering states, i.e., unbounded
sources, possibly with infinite energy, and also
distributional metric solutions to Einstein's equations
\cite{mac6}.
It is interesting to note that a completely different
group theoretic approach to quantum gravity has been developed by Isham \cite{Isham,IshKak0,IshKak}. In this
approach IRS of the infinite dimensional canonical group
$\mathcal C$ for quantum gravity are studied. This fascinating work
also arrives, independently and from another point of
view, at distributional metrics. In Isham's work, however,
these metrics occupy the centre of the stage.



$\mathcal C$ has the semidirect product structure
\be
\mathcal C= \mathcal S \rtimes_{T} G,
\ee
where $\mathcal S$ is an Abelian subgroup of $\mathcal C,$
%infinite dimensional function space,
and $G$ is a \it not \normalfont locally compact group. The
linear representation of $G$ on $\mathcal S$ which specifies
this semidirect product is not relevant here.
Isham does not succeed in obtaining a complete
list of IRS of the canonical group. His efforts in doing so are hindered by two problems:
\begin{enumerate}
\item{The exhaustivity problem: The canonical group is not locally compact,
Mackey's theorems (see e.g. \cite{Mackey1}) ensuring the
% and dealing with the irreducibility
%and
exhaustivity of the IRS constructed by inducing cannot
be invoked, and, as a result, there is no clear idea,
of how many IRS of the canonical group can be obtained
by Wigner$-$Mackey's inducing construction.}
\item{The problem, which proves to be the central mathematical problem in Isham's approach, of
    finding $G$$-$quasi$-$invariant measures on the
    various $G$$-$orbits in  $ \mathcal S^{\prime}$,
    $ \mathcal S^{\prime}$ is the topological dual of $ \mathcal S$ . The actual construction of quasi$-$invariant measures in Isham's approach cannot
    be attained by following the usual method of
   projecting down the Haar measure from $G$ to
   the $G$$-$orbit $\mathcal O \simeq G/L_{f}$,
   $L_{f}$ is the ``little group'' of a representative
   point $f$ of $\mathcal O,$ since $G$ is not locally
   compact and hence has no Haar measure.
   }
\end{enumerate}


The exhaustivity problem in our approach is described in detail in Subsection \ref{extension}  and its solution,
in the Hilbert topology, is given in Subsection \ref{solutions}. One key difference between Isham's approach
and our approach is that in our approach the group $G$ is locally compact. As a result,
%at least in principle,
the method of
   projecting down the Haar measure from $G$ to
   the $G$$-$orbit $\mathcal O \simeq G/L_{f}$ can be invoked, and
$G$$-$quasi$-$invariant measures on the orbits $G/L_{f}$   can be constructed.
For the case of $B(2,1)$ this construction is carried out in Subsection   \ref{invmeas}.


Both problems in Isham's approach have to do with the applicability of Wigner$-$Mackey's induced
representation theory. In our approach in order to apply
Wigner$-$Mackey's theory we need to extend it. As we
stated in the previous Section the required extension
is given in Subsection \ref{extension} and it is reduced to the solution of three problems, and the solution to the three
problems is examined in Subsection \ref{solutions}.













\section{ The group $B(2,1)$ }


\label{sec1}
%Text with citations \cite{RefB} and \cite{RefJ}.
We turn now to the study of $B(2,1),$
%of  to $B(2,1)$,
%of
the analogue of $B$ in three space$-$time dimensions.

\subsection{The group $B^{2,1}(N^{+})$}

\noindent
Recall that the $2+1$
Minkowski space is the vector space $R^{3}$ of row vectors with 3
real components, with the inner product defined as follows. \ Let
$x,y\in R^{3}$ have components $x^{\mu }$ and $y^{\mu }$
respectively, where $\mu =0,1,2$. \ Define the inner product
$x.y$ between $x$ and $y$ by
\begin{equation}
x.y=x^{0}y^{0}-x^{1}y^{1}-x^{2}y^{2}.
\end{equation}






Then the $2+1$ Minkowski space, sometimes written
$R^{2,1}$, is just $R^{3}$ with this inner product.
The ``2,1'' refers to the one plus and two minus signs in the inner product.
Let $SO(2,1)$ be the (connected component of the identity element of the)
group of linear transformations preserving the inner product. \ Matrices
$\Lambda \in SO(2,1)$ are taken as acting by matrix multiplication from the
right, $x\longmapsto x\Lambda ,$ on row vectors $x\in R^{2,1}$. \

The future null cone $ N^{+} \subset R^{2,1}$ is just the set of nonzero vectors
with zero length and $x^{0}>0$:
\begin{equation}
 N^{+}=\left\{ x\in R^{2,1}| x.x=0, x^{0}>0
\right\} .
\end{equation}
Let $R^{*}_{+}$ denote the multiplicative group of all
\it{positive}
\normalfont
real numbers. \ Obviously, if $x\in N^{+}$, then
$tx\in N^{+}$ for any $
t\in R^{* }_{+}$. \ Let $F_{1}(N^{+})$ denote the vector space (under
pointwise addition) of all functions $f:N^{+}\rightarrow R$ satisfying
the homogeneity condition
\begin{equation}
\label{hom}
f(tx)=tf(x)
\end{equation}
for all $x\in N^{+}$ and $t\in R^{*}_{+}$.







Define a representation $T$
of $SO(2,1)$ on $F_{1}(N^{+})$ by setting, for each $x\in N^{+}$ and $\Lambda \in SO(2,1)$,
\begin{equation}
(T(\Lambda )f)(x)=f(x\Lambda ).
\end{equation}
Now let $B^{2,1}(N^{+})$ be the semidirect product
\begin{equation}
\label{tvyrikolasexy}
B^{2,1}(N^{+})=F_{1}(N^{+}) \rtimes_{T}
%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s _{\;T}
SO(2,1).
\end{equation}
That is to say, $B^{2,1}(N^{+})$ is, as a set, just the product
$F_{1}(N^{+})\times
SO(2,1)$, and the group multiplication law for pairs is
\begin{equation}
(f_{1},\Lambda _{1})(f_{2},\Lambda _{2})=(f_{1}+T(\Lambda _{1})f_{2},\Lambda
_{1}\Lambda _{2}).
\end{equation}











Let $R^{*}_{+}$ be the multiplicative group of positive real numbers.
The orbits of the dilatation action
\be R^{*}_{+} \times N^{+} \rightarrow N^{+}; \ (t,x) \mapsto tx,\ee $t \in R^{*}_{+}, \ x \in N^{+},$
are just the open half lines in $N^{+}$ from the origin of $R^{2,1}$. The projective null cone $P( N^{+})$
is precisely the space of orbits of this action.
%these half lines.












We can replace the set of all homogeneous functions, $F_{1}(N^{+})$,
on $N^{+}$, by the set $F(P(N^{+}))$ of all ``free'' functions on $P(N^{+})$, since the homogeneity constraint fixes the
behaviour of functions $f \in F_{1}(  N^{+})$  on these half lines. This gives a bijection
\be
\label{bi1}
F_{1}(N^{+}) \leftrightarrow  F(P(N^{+}))
\ee
by means of which we can find an expression for the representation $T$ in the space $F(P(N^{+}))$, giving us a new realization
\be
\label{grouppr}
B^{2,1}(N^{+})=F(P(N^{+})) \rtimes_{T}SO(2,1).
\ee
%Associate with each open





Each half$-$line  $t  x,$  $ t \in R^{*}_{+}, \ x \in N^{+},$
%the point of intersection of this line with
intersects the 2$-$plane $x^{0}=1$ exactly once; $x^{0}$ is the zero component of $x$. Since $x=(x^{1},x^{2},x^{0}) \in N^{+}$, $x$  is a null vector, therefore, $(x^{1})^{2}+(x^{2})^{2}=(x^{0})^{2}.$
We conclude that the homeomorphic type of the space of orbits  $P(N^{+})$ is a circle, in the model for $P(N^{+})$  constructed here,  given by
\be
(x^{1})^{2}+(x^{2})^{2}=1.
\ee







Rather than work with (\ref{grouppr}), it is more convenient to pass
to a ``spinor'' version of $B^{2,1}(N^{+})$ which double covers this group.




















\subsection{The spinor version of $B^{2,1}(N^{+})$}
%double cover $B^{2,1}(N^{+})_{c}$}
\label{dc}
%$\beta \mathbf{\beta} \bm{\beta} \hm{\beta} %\bm{\beta\mkern-11mu\beta}$
%$\hm{\beta}$
%$\hm{\sigma}$
%$\bm{\sigma}$
%$\bm{\sigma\mkern-11mu\sigma}$
\noindent
%Let ${\rm S}{\rm L}(2,R)$ be the group of all real $2\times 2$ matrices with determinant one.
$
%{\rm
S
%}{\rm
 L
% }
 (2,R)$ is sometimes denoted by $G$ below.
Let ${\rm M}_{s}(2,R)$ be the set of all $2\times 2$ symmetric real matrices.
%Let ${\rm M}(2,R)$ be the set of all $2\times 2$ real matrices,
%and let \be {\rm S}{\rm L}(2,R)=\{ g\in {\rm M}(2,R)|\det g=1 \}.
%\ee ${\rm S}{\rm L}(2,R)$ is a matrix group, sometimes denoted $G$
%below.
We define a right action of $G $ on ${\rm
M}_{s}(2,R)$,
% by
\begin{equation} \label{axcsdfvbgtre}
 {\rm M}_{s}(2,R) \times G \rightarrow {\rm M}_{s}(2,R) \
\rm{with} \  \normalfont (\it{m}\normalfont,\it{g}\normalfont)
\mapsto \it{g}^{\top} \normalfont \it{m}\normalfont \it{g},\normalfont
\end{equation}
where the superscript ${\top}$ means transpose.







Clearly any element $\mu \in {\rm M}_{s}(2,R)$ can be parameterized as follows:
$$
\mu=\left[
\begin{array}{cc}
   x^{\rm o}-x^{1}    &   x^{2} \\
    x^{2}     &  x^{o}+x^{1}
\end{array}  \right]
$$
where $x^{\rm o},x^{1},x^{2} \in R$.
We now consider
the map $b:R^{2,1} \rightarrow {\rm M}_{s}(2,R)$ defined by \begin{equation}
\label{wyujhdcvxsertzawl} b(x)=\left[
\begin{array}{cc}
   x^{\rm o}-x^{1}    &  x^{2} \\
    x^{2}     &  x^{o}+x^{1}
\end{array}  \right],
\end{equation} where the $x^{\mu}$ are the components of $x\in R^{2,1}$.









This
map is a linear bijection, so the right action of $G$ on ${\rm
M}_{s}(2,R)$ induces a linear right action of $G$ on $R^{2,1}$.
The right action of $G$ preserves determinants:
\be
\rm{det}\normalfont(\it{g^{\top} m g})\normalfont=
\rm{det}\normalfont(\it{g}^{\top}) \normalfont
\rm{det}\normalfont(\it{m})\normalfont
\rm{det}\normalfont(\it{g})\normalfont=
\rm{det}\normalfont(\it{m})\normalfont.
\ee
Moreover, \begin{equation} \det(b(x))=x \cdot x. \end{equation}
















Therefore the right action (\ref{axcsdfvbgtre}) of $G $ on ${\rm M}_{s}(2,R)$,
%$$
%{\rm M}_{s}(2,R) \times G \rightarrow {\rm M}_{s}(2,R) \ \rm{with}
%\ \normalfont (\it{m}\normalfont,\it{g}\normalfont)
%\mapsto \it{g}^{\top} \normalfont \it{m}\normalfont %\it{g}\normalfont,
%$$
corresponds, via the linear bijection $R^{2,1} \leftrightarrow {\rm M}_{s}(2,R)$, to linear maps preserving lengths $x \cdot x.$ In fact, all linear maps $ \Lambda \in SO(2,1)$ are obtained in this way, and the action (\ref{axcsdfvbgtre}) gives a homomorphism
\begin{equation} \gamma : G
\rightarrow SO(2,1),
%\quad {\rm with}
\end{equation}
which is onto, and has kernel $ Z_{2}=\{ Id,-Id \} $ in
$G$, $Id$ denoting the identity element of $G$.
Thus $ \gamma $ identifies $ G  $ as the
double cover of $ SO(2,1), $ \begin{equation} G = SO(2,1)_{c}. \end{equation}













The null cone $N^{+}$ becomes, under the identification
$R^{2,1} \leftrightarrow {\rm M}_{s}(2,R), $
$\widetilde{N^{+}}=b(N^{+})$ with
%We can identify $R^{2,1}$ with the set
%${\rm M}_{s}(2,R)$ of all symmetric real $2\times 2$ matrices by %means of the
%bijection $b$ defined above.
%The null cone $N^{+}$ becomes, under this
%identification
\begin{equation}
\widetilde{N^{+}}
=\left\{ \mu \in
{\rm M}_{s}(2,R),\;\mid x^{0} > 0,   \;
\det \mu=0\right\} .
\end{equation}
Thus $\mu \in \widetilde{N^{+}}
$ if and only
if $\mu $ has rank exactly $1$ and $x^{0} > 0$.
Let $\scri$ be the set of all non$-$zero real two$-$component row vectors $\bm{\sigma}$; $\scri=R^{2}- \{0\}$.
From the rank condition and the requirement $x^{0} > 0$, it follows that
\begin{equation}
\label{sp}
\mu \in \widetilde{N^{+}}
\quad \Leftrightarrow \quad \mu =\bm{\sigma}^{\top}\bm{\sigma}
\end{equation}
where  $\bm{\sigma} \in \scri$.
\noindent














Equation (\ref{sp}) gives a parametrization of
the null cone $
\widetilde{ N^{+}}
$ by
means  of spinors.
However, this parametrisation, though (by construction) onto, is not one$-$one. It is onto because \it{every}  \normalfont
$\mu \in \widetilde{N^{+}}$   can be parameterized in this form but it is not one$-$one because both $\bm{\sigma} $ and $-\bm{\sigma} $ give rise to the same $\mu \in \widetilde{N^{+}}.$

%\noindent





There is a natural   left action of $R^{*}_{+}$ on $\widetilde{ N^{+}}$
\be R^{*}_{+} \times \widetilde{ N^{+}} \rightarrow \widetilde{ N^{+}}; \ (t,\mu) \mapsto t\mu, \ee
$t \in R^{*}_{+}, \ \mu \in \widetilde{ N^{+}}.$
The aforementioned linear bijection
$$
b: N^{+} \rightarrow \widetilde{ N^{+}}  \
%\rm{with} \  \normalfont
$$
is preserved by the left $R^{*}_{+}$ action on $N^{+}$
and on $\widetilde{ N^{+}}.$








Therefore the orbits of the $R^{*}_{+}$ action on $N^{+}$
are in bijective correspondence with the orbits of the $R^{*}_{+}$ action on $\widetilde{ N^{+}}.$ Consequently, with this identification, we can use as a model for the
orbits of the $R^{*}_{+}$ action on $N^{+}$, the
orbits of the $R^{*}_{+}$ action on $\widetilde{ N^{+}}$.
Let $\mathcal O=t \mu,$ $t \in R^{*}_{+},\ \mu \in \widetilde{N^{+}},$
be an orbit of the $R^{*}_{+}$ action on $\widetilde{N^{+}}.$
Let $\mathcal O_{r}$ be any representative of this orbit.
$\mathcal O_{r}$ can be parameterized in terms of
spinors, and, as (\ref{sp}) indicates,
if $\bm{\sigma} $  corresponds to $\mathcal O_{r}$ then $ -\bm{\sigma} $ also corresponds to $\mathcal O_{r}.$








We conclude that the projective space $P(\widetilde{ N^{+}})$ of the orbits of the $R^{*}_{+}$ action on $\widetilde{N^{+}}$ is
\be
P(\widetilde{ N^{+}})=P_{1}(R)= S^{1}/Z_{2} \simeq S^{1},
\ee
%$P_{1}(R) $ is the one$-$dimensional real projective  space (the circle quotient the antipodal map).
The fact that both $\bm{\sigma}$ and $-\bm{\sigma}$ correspond to
the \it same \normalfont orbit $\mathcal O_{r}$ provides the starting point for a more rigorous proof \cite{Melas}
of the homeomorphic type of  $P(\widetilde{ N^{+}}).$
%A sounder proof of the homeomorphic type of  $P(\widetilde{ N^{+}})$ %is given in \cite{Melas}.
%This proof for the homeomorphic type of  $P(\widetilde{ N^{+}})$ can %be given \cite{Melas} on a sounder basis.

%\noindent
%We conclude that
Consequently the spinor version $B^{2,1}(N^{+})_{c}$
of $B^{2,1}(N^{+})$ which double covers $B^{2,1}(N^{+})$ is
\be
B^{2,1}(N^{+})_{c}=F(P(\widetilde{ N^{+}}) \rtimes_{T}G,
\ee
where $G=SL(2,R).$ Interestingly enough
$F(P(\widetilde{ N^{+}})$ can be realized in two different ways
which give rise to two different isomorphic realizations of
$B^{2,1}(N^{+})_{c}.$

\vspace{0.2cm}
\noindent
$\bf 1^{st} \ realization.$ \normalfont \hspace{0.2cm}
$F(P(\widetilde{ N^{+}})$ is the space  of ``free'' real$-$valued
functions $F(P_{1}(R))$ defined on $P_{1}(R)$.
Thus in this case $B^{2,1}(N^{+})_{c}$ is realized as
\be
B^{2,1}(N^{+})_{c}=F(P_{1}(R)) \rtimes_{T}G.
\ee

\vspace{0.2cm}
\noindent
$\bf 2^{nd} \ realization.$ \normalfont \hspace{0.2cm}
Let $S^{1}$ be the curve $S^{1} \subset I$ defined by
\be
S^{1}=\{ \bm{m}  \in I | \ |\bm{m}|=1\}.
\ee
$F(P(\widetilde{ N^{+}})$ is the space  of ``even'' real$-$valued
$\beta:S^{1} \rightarrow R.$ That is, these functions satisfy the
even$-$ness condition
\be
\beta(\bm{m})=\beta(-\bm{m}).
\ee
We denote this space by $F_{e}(S^{1}).$
Thus in this case $B^{2,1}(N^{+})_{c}$ is realized as
\be
B_{e}^{2,1}(N^{+})_{c}=F_{e}(S^{1}) \rtimes_{T}G.
\ee
%where the subscript $e$ differentiates the second realization from %the first.











\vspace{0.2cm}
%\noindent






We can use either of these realizations to develop the representation theory of $B^{2,1}(N^{+})$.
The representation theory which has as point of departure the
first realization cannot be mapped onto the representation theory  which has as point of departure  the second realization. Each representation theory presents its own difficulties, problems and  subtleties. However, as expected, the results of the representation theory which stems from the first realization can be naturally mapped onto the results of the representation theory which stems
from the second realization. Here we restrict attention to the representation theory which originates from the first realization.









%established before is preserved by the dilatation action
%Thus the orbits are in bijective correspondence with the orbits and %therefore their homeomorphic type is the same.
%We define a right action of $G $ on ${\rm
%M}_{s}(2,R)$,
%% by
%\begin{equation} \label{axcsdfvbgtre}
% {\rm M}_{s}(2,R) \times G \rightarrow {\rm M}_{s}(2,R) \
%\rm{with} \  \normalfont (\it{m}\normalfont,\it{g}\normalfont)
%\mapsto \it{g}^{\top} \normalfont \it{m}\normalfont %\it{g},\normalfont
%\end{equation}
























%Therefore, the double cover of the group $B^{2,1}(N^{+})$,
%given in (\ref{tvyrikolasexy}), has the form
%\begin{equation} B^{2,1}(N^{+})_{c}=
%F_{1}(N^{+}) \rtimes_{T} {\rm S}{\rm L}(2,R).
%%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T} {\rm S}{\rm L}(2,R).
%\end{equation}
%Strictly speaking, $``T"$ should
%read $``T\gamma"$, but the notation is simpler as above.

\subsection{The group $B(2,1)$}
\label{sec2.3}


\noindent
In the $1^{\rm{st}}$ realization $F(P(\widetilde{ N^{+}})$ is realized as
the space  $F(P_{1}(R))$ of ``free'' real$-$valued
functions  defined on $P_{1}(R)$.
As  a result we have the following Theorem:
%$B^{2,1}(N^{+})_{c}$
%is realized as follows:
%In this realization $B^{2,1}(N^{+})_{c}$ is
%described in the following Theorem.





%It is a key result \cite{Melas}, which leaves its imprint
%on representation theory, that
%$$
%P(N^{+})=R^{*}_{+} \backslash N^{+} \simeq P_{1}(R),
%%\simeq S^{1}
%$$
%$P_{1}(R) \equiv S^{1}/Z_{2}$ is the one$-$dimensional real %projective space (the circle quotient the antipodal map); $P_{1}(R) %\simeq S^{1}.$
%The group $B^{2,1}(N^{+})_{c}$ can be realized \cite{Melas} in %terms of
%$F(P(N^{+}))$, i.e., in terms of ``free'' real$-$valued
%%i.e. unconstrained, real$-$valued
%functions defined on $P_{1}(R)$. In particular the following holds
%\cite{Melas}:
\begin{theorem}
The group $B^{2,1}(N^{+})_{c}$ can be realized as
\be
\label{defb}
B^{2,1}(N^{+})_{c}=F(P_{1}(R)) \rtimes_{T} G,
%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T} G
\ee
with semidirect product specified by
\be
\label{actb}
(T(g)\alpha)(\sigma)=\kappa_{g}(\sigma)\alpha(\sigma g).
\ee
\end{theorem}

\vspace{0.2cm}

\noindent
$G$ is the group SL(2,$R$), $g \in G$,
$\alpha \in F(P_{1}(R))$.
Moreover, if
\begin{equation}
g=\left[ \begin{array}{lr}
   a    & b \\
   c    & d
\end{array}  \right]
,
\end{equation} then the components $ \sigma_{1},\sigma_{2} $ of $\bm{\sigma} \in R^{2}-\{0 \}$ transform
linearly under $g$, so that the ratio $ \sigma= {\sigma_{1}}/{\sigma_{2}}, \ \sigma_{2} \neq 0 $, \  transforms
fraction linearly under $g$. Writing $ \sigma
g $ \ for the transformed ratio, \begin{equation}
\label{aliki} \sigma g=
\frac {(\bm{\sigma}g)_{1}}{(\bm{\sigma}g)_{2}}=\frac
{\sigma_{1}a + \sigma_{2}c} {\sigma_{1}b+\sigma_{2}d}=
\frac { \sigma a+c}{\sigma   b+d}. \end{equation}
The ratio $ \sigma= {\sigma_{1}}/{\sigma_{2}}, \ \sigma_{2} \neq 0 $, is a local inhomogeneous coordinate of $P_{1}(R)$.
The factor $\kappa_{g}(\sigma)$ on the right hand side of (\ref{defb})
turns out \cite{Melas} to be
\begin{equation}
\label{donkey}
\kappa_{g}(\sigma)=
\frac {(\sigma b+d)^{2}+(\sigma a+c)^{2}}{1+\sigma^{2}}.
\end{equation}










In analogy to $B$, it is natural to choose a measure $\lambda$
on $P_{1}(R)$ which is invariant under
the maximal compact subgroup $ S O (2)$ of $ G $; so we choose $\lambda$ to be the standard
normalized Lesbegue measure $ {\rm d} \lambda = \frac{{\rm d} \theta}{2 \pi}$.
%So far, the supertranslation space $F_{1}(N^{+})$ has been defined as a %space of truly \it arbitrary \normalfont
%homogeneous  functions of degree one. This has been merely for clarity;
For physical applications, it is necessary to give $F(P_{1}(R))$
%this space
additional structure. For reasons discussed in detail in McCarthy \cite{mac1},
the supertranslation space $F(P_{1}(R))$  is restricted
to be the separable Hilbert space $L^{2}(P_{1}(R),\lambda ,R)$
of real$-$valued functions defined on
$P_{1}(R) \simeq S^{1}$; functions
square integrable
with respect to $\lambda$.
%the standard normalized (Lesbegue) measure $\lambda$
%on $P_{1}(R) \simeq S^{1}$.
$B^{2,1}(N^{+})_{c}$ can now be realized in terms of $L^{2}(P_{1}(R),\lambda ,R)$ \cite{Melas} as follows:
%For reasons discussed in detail in McCarthy \cite{mac1}, we now give a %new realization of $B^{2,1}(N^{+})_{c}$
%where the supertranslation space is restricted
%to be the separable Hilbert space $L^{2}(P_{1}(R),\lambda ,R)$
%of real$-$valued functions defined on
%$P_{1}(R) \simeq S^{1}$; functions
%square integrable
%with respect to
%the standard normalized (Lesbegue) measure $\lambda$
%on $P_{1}(R) \simeq S^{1}$.
% %$P_{1}(R) \equiv S^{1}/Z_{2}$ is the one$-$dimensional real projective %% space (the circle quotient the antipodal map).

%In particular in \cite{Melas}
%the following Theorem is proved:
\begin{theorem}
\label{omorpho} The group $B^{2,1}(N^{+})_{c}
$   can be realised as \begin{equation}
B(2,1)
= L^{2}(P_{1}(R),\lambda ,R)
\rtimes_{T} G \end{equation} with semidirect
product specified, as before, by
%\begin{equation}
$$
\label{an}
(T(g)\alpha)(\sigma)=\kappa_{g}(\sigma)\alpha(\sigma g).
$$
%\end{equation}
\end{theorem}
\vspace{0.2cm}

\noindent
In the last relation
%function $\alpha$ in the last relation belongs to
$\alpha \in L^{2}(P_{1}(R),\lambda ,R),$ and $\kappa_{g}(\sigma)$
is given, as before,
%again
by (\ref{donkey}).
We denote our final realization of our group
by $B(2,1)$ to
distinguish it from the previous realizations $B^{2,1}(N^{+})$ and $B^{2,1}(N^{+})_{c}$.






%\noindent
Reisz$-$Fr$\acute{e}$chet theorem for Hilbert spaces \cite{Rudin}
%we know that
states   that the topological dual of a Hilbert space can be identified
with the Hilbert space itself, so that we
have $ {L^{2}}^{'}(P_{1}(R),\lambda ,R) \simeq
L^{2}(P_{1}(R),\lambda ,R).$
In particular, given a continuous linear functional
 $f \in {L^{2}}^{'}(P_{1}(R),\lambda ,  R)$, we can write,
 for $\alpha \in L^{2}(P_{1}(R),\lambda ,R)$
\begin{equation}
(f,\alpha)=<f,\alpha>,
\end{equation}
where the function $f \in L^{2}(P_{1}(R),\lambda ,R)$ on the right is uniquely determined
by (and denoted by the same symbol as) the linear functional $f \in {L^{2}}^{'}(P_{1}(R),\lambda ,R)
$ on the left.








The representation theory of
$B(2,1)$
is governed by the dual
action $T'$ of $ G$ on the topological dual
$ {L^{2}}^{'}(P_{1}(R),\lambda ,R)$ of $L^{2}(P_{1}(R),\lambda ,R).$
The dual action $T'$ is defined by:
\begin{equation} <T'(g)f,\alpha>=<f,T(g^{-1})\alpha>\cdot
\end{equation}
A
short calculation
gives
\begin{equation} \label{tzatziki}
(T'(g)f)(\sigma)=\kappa_{g}^{-2}(\sigma)f(\sigma g).
\end{equation}
Now, this action $T'$ of $ G$
on ${L^{2}}^{'}(P_{1}(R),\lambda ,R)$,
given explicitly above is, like the action $T$ of
$ G$ on $L^{2}(P_{1}(R),\lambda ,R)$, continuous. The ``little group'' $L_{f}$
of any  $f \in {L^{2}}^{'}(P_{1}(R),\lambda ,R)$ is the stabilizer \begin{equation}
L_{f}=\{g \in  G
\ | \ T'(g)f=f\}. \end{equation}

%By continuity, $L_{\phi}\subset  G$ is a closed subgroup.


\section{Wigner$-$Mackey's theory}
\label{sec2}








Our construction of the IRs of $B(2,1)$ is based on G.W.Mackey's pioneering work on group representations (see, for example, \cite{Mackey0,Mackey00,Mackey,Mackey1});
%Mackey2
%});
in particular, is based on an extension to the relevant infinite dimensional case
\cite{mac6,Mel5}
%McCMel,Mel5}
of his semidirect product theory.
One of the sources of Mackey's  work
%It is interesting that the main sources of Mackey's  work %were his extension  of the Stone$-$von Neumann theorem %\cite{Stone}, which is the
%cornerstone of quantum mechanics, Von Neuman's theory of %direct integral decompositions
%of operator algebras which he developed in the 1930s,
%and
was Wigner's classic paper \cite{Wigner} in which he broke new grounds in mathematics by giving the first explicit treatment of of infinite dimensional
representations of a Lie group.











We will give the bare essentials of
Wigner$-$Mackey's theory  in order to construct explicitly
the operators of the induced representations of $B(2,1).
%\mathcal B \mathcal B_{c}
$ We will not follow the functional analytic treatment of induced
representations but instead we shall adopt the more geometrical
fibre bundle dominated scheme. Standard references are
(\cite{Wigner},\cite{Mackey},\cite{Mackey1},\cite{Simms},
\cite{Isham} and references therein).



Let $ A $ and $ \mathcal G $ be topological groups, and let $ T $
be a given homomorphism from $ \mathcal G $ into the group of
automorphisms ${\rm Aut} (A)$ of $A.$ Suppose $A$
is abelian and
$\mathcal B = A  \rtimes_{T} \mathcal G $
%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T}
%\mathcal G $
is the semidirect product of $A$ and $ \mathcal G,$
specified by the continuous action $T: \mathcal G
\longrightarrow  {\rm Aut} (A).$ In the product topology of
$A \times \mathcal G,$ $\mathcal B $ then becomes a topological
group. It is assumed that it becomes a separable locally compact topological
group.


The irreducible unitary continuous representations of $A$ (characters) %can be
are
\noindent
%A \it{character} \normalfont $\chi$ of $\mathcal A$ is a
continuous homomorphisms
\be
\phi:  A \rightarrow U(1)
\ee
of $ A$ into the multiplicative group of complex numbers of unit modulus. A composition law
\be
\label{com}
(\phi_{1} \phi_{2}) (\alpha) = \phi_{1}(\alpha) \phi_{2}(\alpha)
\ee
on the characters and a multiplication by scalars
\be
(\kappa \phi)(\alpha)=\kappa \phi(\alpha), \ \kappa \in R,
\ee
turns the characters into a vector space over the reals.
%given the structure of an abelian group $\hat{A},$ the dual group of
%$A.$
In particular the composition law (\ref{com}) turns the characters
into an abelian group, the dual group $\hat{A}$ of $A$.



Writing the character $\phi$ as
\be
\label{chi}
\phi(\alpha)=e^{i f(\alpha)},
\ee
one easily finds \cite{Melas}
that the real$-$valued function $f$
%\begin{enumerate}
%\item Is continuous,
%\item Is additive, i.e., %$f(\alpha_{1}+\alpha_{2})=f(\alpha_{1})+f(\alpha_{2})$, and,
%\item $f(\kappa  \alpha) = \kappa f(\alpha), \ \kappa \in R.$
%\end{enumerate}
%Therefore
%$f$
is a continuous linear functional on $ A$, that is, $f$ belongs to the topological dual $ A^{\prime}$ of $ A$. The map
\begin{eqnarray}
j:  A^{\prime} & \rightarrow & \hat{ A}, \ \rm{given \ by} \\
\label{isoj}
j(f) & = & e^{i f}
\end{eqnarray}
provides an isomorphism between $\hat{ A}$ and $ A^{\prime}$, as real vector spaces, and in particular as abelian groups.
We shall identify, if required, the sets
$\hat{ A}$ and $ A^{\prime}$.






The action $T$ of $\mathcal G $ on $ A $
induces a dual action on $ \hat{A}.$
Indeed, any bijective map $b: A \longrightarrow A$,
induces a map $\hat{A} \longrightarrow  \hat{A}$,
$\phi  \longmapsto b \phi$, defined by
\be
\label{indact}
(b \phi)(a)=\phi(b^{-1}a).
\ee
It is to be noted that the bijective correspondence
%$A \leftrightarrow  A^{\prime}$
\be
\label{bc}
\hat{ A} \leftrightarrow  A^{\prime}
\ee
is preserved pointwise by the $\mathcal G$ action
if and only if the action of $\mathcal G$
on $ A^{\prime}$ is defined by
\be
\label{da}
(gf)(\alpha)=f(g^{-1}\alpha).
\ee
%The action $T$ of $\mathcal G $ on $ A $
%induces a dual action on $ \hat{A}.$
%Indeed, any bijective map $b: A \longrightarrow A$,
%induces a map $\hat{A} \longrightarrow  \hat{A}$,
%$\phi  \longmapsto b \phi$, defined by
%\be
%\label{indact}
%(b \phi)(a)=\phi(b^{-1}a).
%\ee
In this way an action $\alpha \longmapsto  h   \alpha h^{-1}   $      of $\mathcal B$ on $A$ induces an action
$\phi \longmapsto h \phi$ of $\mathcal B$ on $\hat{A}$,
\be
\label{actionaction}
(h \phi)(\alpha)= \phi (h^{-1} \alpha h).
\ee
\noindent








For a  given character
$ \phi \in \hat{A},$ the largest subgroup $L_{\phi} $
of $\mathcal G $ which leaves $\phi $ fixed is called
the ''little group'' of $ \phi $, and the set of characters which can be
reached from $ \phi $ by the
$  \mathcal B $ action is called the orbit
of $ \phi $, denoted $ \mathcal B \phi.$
The largest subgroup of $ \mathcal B $ which leaves $\phi $
fixed is the semidirect product
$ \mathcal B_{\phi}=
A \rtimes_{T} L_{\phi }  \;$.
%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T} L_{\phi %}  \;$.
$\mathcal B_{\phi} $ is a closed subgroup of
$ \mathcal B. $

Let U be a continuous irreducible unitary representation
of the ''little group''
$ L_{\phi} $ on a Hilbert space $ D. $
Then
\be
\phi U:  (a,l) \longmapsto \phi(a) U (l),
\ee
where $a \in A $ and $ l \in L_{\phi}, $
is a continuous unitary irreducible representation of the group
$ \mathcal B_{\phi}=
A \rtimes_{T}
%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T}
L_{\phi } $
in $ D.$

Let $ \mathcal B_{\phi} \rightarrow \mathcal B \rightarrow
\mathcal B / \mathcal B_{\phi} $ be the principal fibre bundle
with fibre $\mathcal B_{\phi},$ base space the coset space
$ \mathcal B / \mathcal B_{\phi} $ and the smooth projection
map $ \pi : \mathcal B \rightarrow \mathcal B / \mathcal B_{\phi} $ is the usual function $ \pi(h):= h \mathcal B_{\phi} $ which associates with each $ h \in  \mathcal B  $
the coset $ h \mathcal B_{\phi} $ to which it belongs. $\;
\mathcal B_{\phi}$  acts upon $ D $  (on the left) via
%as a
%group of diffeomorphisms, via
\begin{eqnarray}
\phi U &  :  & \mathcal B_{\phi}  \longrightarrow {\rm Aut}(D), \nonumber \\
&& (a,l) \longmapsto \phi(a) U(l).
\end{eqnarray}




There is a right action of $ \mathcal B_{\phi} $ on $\mathcal B
\times D $ defined by
\begin{equation}
\label{kguhv,hliytgl7ytgyukltgol7g}
(h,l)h_{\phi} \quad := \quad (hh_{\phi},h^{-1}_{\phi}v), \quad h_{\phi} \in
\mathcal B_{\phi}.
\end{equation}
The bundle with fibre $D, $
associated to the principal bundle
$ \mathcal B_{\phi} \rightarrow \mathcal B \rightarrow
\mathcal B / \mathcal B_{\phi},$
has
%the
total space
$ \mathcal B \times_{{\mathcal B}_{\phi}}  D,$ %\;$,
which is defined to be the orbit space of $ \mathcal B \times D $
under the action (\ref{kguhv,hliytgl7ytgyukltgol7g}), and
projection map
\begin{eqnarray}
\pi_{D}  &  :  & \mathcal B \times_{\mathcal B_{\phi}} D
\longrightarrow \mathcal B/ \mathcal B_{\phi}, \nonumber \\
&& [h,v] \longmapsto \pi(h),
\end{eqnarray}
where $ [h,v] $
denotes the equivalence class of $ (h,v) \in \mathcal B \times D $
under (\ref{kguhv,hliytgl7ytgyukltgol7g}).













That $ \pi^{-1}_{D} (h \mathcal B _{\phi}) $
really is a copy $D_{h{\mathcal B}_{\phi}} $ of $ D $
can be seen by noting the existence of the map
\begin{eqnarray}
i  &  :  & D  \longrightarrow D_{\pi(h)}, \nonumber \\
&& v \longmapsto [h,v],
\end{eqnarray}
which is a diffeomorphism for each $ h \in \mathcal B. $



Therefore, for {\it each} (continuous unitary irreducible)
representation $U$ of the ''little group'' $L_{\phi}$
and for {\it each} character $ \phi $ we
have a bundle
$ D \rightarrow \mathcal B \times_{{\mathcal B}_{\phi}}  D
\rightarrow  \mathcal B / \mathcal B_{\phi}$
with fibre $ D,$
total space
$ \mathcal B \times_{{\mathcal B}_{\phi}}  D,$
and base space
$\mathcal B / \mathcal B_{\phi}.$
The bundle
$ D \rightarrow \mathcal B \times_{{\mathcal B}_{\phi}}  D
\rightarrow  \mathcal B / \mathcal B_{\phi}$
depends, up to $ \mathcal B $$-$isomorphism, only on the orbit
$\mathcal B \phi
%= \mathcal G \phi
$
and not on the choice
of $ \phi. $ There is a natural bijection
\be
\mathcal B / \mathcal B_{\phi} \longrightarrow \mathcal B \phi,
\ee
$h \mathcal B_{\phi} \longrightarrow h \phi$ which will even be a homeomorphism \cite{Bour} (Appendix I), under quite general conditions.
 We
 will
 %in  future
 often identify $\mathcal B / \mathcal B_{\phi}$ and $\mathcal B \phi$.

Since $A$  acts trivially on $\hat{A}$
the orbit $\mathcal B \phi$ is identical to the
orbit $\mathcal G \phi$.
Moreover,  the map
\be
\mathcal G/L_{\phi} \longrightarrow \mathcal B \phi, \quad g  L_{\phi} \longmapsto h \phi,
\ee
where $g \in \mathcal G,$ and $h=(\alpha, g), \ \alpha \in A,$ is
also a bijection.
% a bijection as well.
For this reason, we shall also identify the sets $\mathcal G/L_{\phi}$ and $\mathcal B \phi$. To conclude, the following bijections hold
%re is a natural bijection between any two of the %following sets
\be
\label{omg}
\mathcal B / H_{\phi} \longleftrightarrow \mathcal B \phi
 \longleftrightarrow \mathcal G/L_{\phi} \longleftrightarrow \mathcal G \phi,
\ee
and, for this reason, we shall identify any two of
the sets $\mathcal B / H_{\phi}, \ \mathcal B \phi, \ \mathcal G/L_{\phi}$, and $\mathcal G \phi$.
%them.











It is well known (see e.g. \cite{Simms}, p.45) that
when $\mathcal G$ is locally compact and $L_{\phi}$
is a closed subgroup of $\mathcal G$, then there is
a unique non$-$zero invariant measure class M on
$\mathcal G/L_{\phi}$. For each measure $\mu$
in M and each $g \in \mathcal G$ the measure $\mu_{g}$:
\be
\mu_{g}(E)=\mu(g^{-1}E),
\ee
where $E$ is a Borel subset of $\mathcal G/L_{\phi}$, is also in $M$.
%Let $M$ be the unique invariant measure class on
%$ \mathcal B / \mathcal B_{\phi}.$
% = \mathcal G / L_{\phi} \;$.






For each $ \mu \in M $ let
\begin{eqnarray}
H_{\mu} &  = &  \left  \{
\psi \mid \psi
\quad {\rm a} \quad {\rm Borel} \quad {\rm section}
\quad {\rm of} \quad
D \rightarrow \mathcal B \times_{{\mathcal B}_{\phi}}  D
\rightarrow  \mathcal B / \mathcal B_{\phi}, \quad {\rm and}
\right . \nonumber \\
&& \left .
\quad
\int_{{\mathcal G}/L_{\phi}}<\psi(p),\psi(p)>{\rm d} \mu (p) < \infty
\right  \},
\end{eqnarray}
where $ p \in {\mathcal G}/L_{\phi} $ and $ <\psi(p),\psi(p)>$
denotes the inner product in the Hilbert space $\pi_{D}^{-1}(p)=D_{p}$
which is diffeomorphic to $ D $.
With the inner product
\be
< \psi , \phi > = \int_{ \mathcal G / L_{\phi}}
< \psi(p) , \phi(p) > {\rm d} \mu(p),
\ee
and with sections identified if they differ only on a set of $ \mu$$-$measure
zero, $H_{\mu}$ is a Hilbert space.








Define an action of $\; \mathcal B \;$ on $\; H_{\mu} \;$ by
\be
\label{ukygvkhvkyutfkyufvmhgv}
(h \psi )(p)=
\sqrt{ \frac {{\rm d} \mu_{h}}{{\rm d} \mu}(p)} h( \psi (h^{-1}p)),
\ee
where $ h \in \mathcal B,  p \in \mathcal B / \mathcal B_{\phi},$ and $\; \mu_{h}(E)= \mu(h^{-1} E) \;$ for any Borel subset $\; E \;$
of $\; \mathcal B / \mathcal B_{\phi}. \;$ This action is a unitary action.
The representation of $ \mathcal B $ on  $ H_{\mu} $
given by (\ref{ukygvkhvkyutfkyufvmhgv}) is called the representation of
$ \mathcal B $ induced by the representation $ U $ of $ L_{\phi}$. If $ U $ and $ W $ are equivalent representations of
$ L_{\phi} $ then they yield equivalent induced representations of
$ \mathcal B $.


If it is further assumed \cite{Mackey1}
that $\mathcal B$
is a \it{regular} \normalfont semidirect product then
\it{all} \normalfont IRS of $ \mathcal B $
are obtained by inducing from the IRS of
the ``little groups''
%are induced
in
the way described above.
%The $ \mathcal B $  action
%(\ref{actionaction}) on $ \hat A $  turns $ \hat A $ into
%a $ \mathcal B-$orbit$-$space.
Regular
means that the $ \mathcal B-$orbit$-$space $ \hat A $
%this orbit space
is nice in a measure$-$theoretic
sense: The $ \mathcal B-$orbits can be enumerated in some way,
namely, there is a Borel set in $ \hat A $ that meets each
$ \mathcal B-$orbit
exactly once.



\vspace{0.1cm}

The central results of induced representation theory are the following:
\begin{enumerate}
\item
{Given the topological restrictions on
$\mathcal B=
A \rtimes_{T} \mathcal G $
%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T}
%\mathcal G $
(separability and local compactness), any representation of $ \mathcal B,$
constructed by the method above,  is irreducible iff the representation
$ U $ of $ L_{\phi} $ on $ D$ is irreducible.
%Thus
An
irreducible representation of $\mathcal B $ is obtained for {\it each}
$ \phi \in \hat A $ and {\it each} irreducible representation $ U $
of $L_{\phi}$.}
\item
{If $ \mathcal B=
A \rtimes_{T} \mathcal G $
%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T}
%\mathcal G $
is a regular semidirect product
%(i.e., $\hat A $ contains a Borel subset which meets each %orbit
%in $ \hat A $ under $ \mathcal B $ in just one point)
then {\it all} of its IRS
%irreducible
%representations
can be obtained in this way.}
\end{enumerate}





It follows from the previous discussion that in order to give
the operators of the IRS of $B(2,1)$  explicitly
%induced representations explicitly
it is
necessary to give the following information
\begin{enumerate}
\item{
An irreducible unitary representation $U$ of $L_{\phi}$ on a
Hilbert space $D$ for each $L_{\phi}.$}
\item{
A $ \mathcal G$$-$quasi$-$invariant measure $ \mu $
on each orbit $ \mathcal G \phi
\approx \mathcal G / L_{\phi};$
where $L_{\phi} $ denotes the ''little group'' of the base point
$ \phi \in \hat A $ of the orbit
$ \mathcal G \phi.$
}
\end{enumerate}
To find the IRS
%induced representations
of
$ B(2,1)
= L^{2}(P_{1}(R),\lambda ,R)
\rtimes_{T} G$
%\mathcal B \mathcal B_{c}=
%A( \mathcal N )   \bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T} %\mathcal G, $
then, it is enough to provide the information cited in 1 and 2
for each of the orbit types.









%Let $\; A \;$ and $\; \mathcal G \;$ be topological %groups, and
%let $\; T \;$ be a given homomorphism from $\; \mathcal G %\;$ into
%the group of automorphisms $\; {\rm Aut} (A) \;$ of $\; %A. $
%Suppose $\; A \;$ is abelian and $\; \mathcal B= A
%\rtimes_{T} \mathcal G \;$ is the
%semidirect product of $\; A \;$ and $\; \mathcal G, $ %specified
%by the continuous action $\; T \; : \mathcal G %\longrightarrow
%{\rm Aut} (A). \;$ In the product topology of $\; A %\times
%\mathcal G \;$, $\; \mathcal B \;$ then becomes a %topological
%group. It is assumed that it becomes a separable locally %compact
%topological group.
%\noindent
% In order to give the operators of the induced
%representations explicitly it is necessary
%(\cite{Wigner}, \cite{Mackey}, \cite{Mackey1}, %\cite{Simms},
% \cite{Isham} and references therein) to give %the following
%information:
%\begin{enumerate}
%\item{ An irreducible unitary representation %$\;U\;$ of
%$\;L_{\phi}\;$ on a Hilbert space $\;D\;$ for %each
%$\;L_{\phi}.\;$} \item{ A $\; \mathcal %G\;$$-$quasi$-$invariant
%measure $\; \mu \;$ on each orbit $\; \mathcal %G \phi \approx
%\mathcal G / L_{\phi}, \;$ where $\; L_{\phi} %\;$ denotes the
%''little group'' of the base point $\; \phi \in
%%\hat
%A^{'} \;$ of the
%orbit $\; \mathcal G \phi; $ $A^{'}$ is the %topological dual of $A$. }
%\end{enumerate}


%\noindent Let $\; D_{\mu} \;$ be the space of %functions $\; \psi:
%\mathcal G \rightarrow D \;$ which satisfy the %conditions



%\begin{eqnarray}
%(a) & \psi(gl)=U(l^{-1})\psi(g), & g \in %\mathcal G, \ l \in
%L_{\phi}, \nonumber \\
%& & \nonumber \\ (b) & \int_{\mathcal {G} %\phi}
%<\psi(q),\psi(q)>{\rm d}\mu(q) < \infty,  %\nonumber
%\end{eqnarray}
%where the scalar product under the integral %sign is that of $\; D.
%\;$ Note, that the constraint $\; (a) \;$ %implies that $\;
%<\psi(gl),\psi(gl)>=<\psi(g),\psi(g)>, \;$ and %therefore the inner
%product $\;<\psi(g),\psi(g)>,$ $\ g \in %\mathcal G, \;$ is constant
%along  every element $\; q \;$ of the coset %space $\; \mathcal
%G/L_{\phi} \approx \mathcal G \phi.$ This %allows to assign a
%meaning to $\; <\psi(q),\psi(q)>, $  where $\; %q=gL_{\phi}, \;$ by
%defining $\; <\psi(q),\psi(q)>:= %<\psi(g),\psi(g)>. \;$ Thus the
%integrand in $\; (b) \;$ becomes meaningful due %to the condition
%$\; (a). \;$ A pre$-$Hilbert space structure %can now be given to $\;
%D_{\mu} \;$ by defining the scalar product
%\begin{equation}
%\label{hujkiolokijuhy} %<\psi_{1},\psi_{2}>=\int_{\mathcal G
%\phi}<\psi_{1}(q),\psi_{2}(q)> {\rm d}\mu(q),
%\end{equation}
%where $\; \psi_{1},\psi_{2} \in D_{\mu}. \;$ It %is convenient to
%complete the space $\; D_{\mu} \;$ with respect %to the norm
%defined by the scalar product %(\ref{hujkiolokijuhy}). In the
%resulting Hilbert space, functions are %identified whenever they
%differ, at most, on a set of $\; \mu-\!$measure %zero. Thus our
%Hilbert space is
%\begin{equation}
% D_{\mu}=L^{2}(\mathcal G \phi, \mu ,D).
%\end{equation}
%Define now an action of $\; \mathcal B= A
%\rtimes_{T} \mathcal G \;$ on $\;
%D_{\mu}\;$ by
%\begin{eqnarray}
%\label{e1}
%(g_{{\rm o}}\psi)(q) & = & %\sqrt{\frac{\rm{d}\mu_{g_{{\rm %o}}}}{\rm{d}\mu}(q)} \psi(g_{{\rm o}}^{-1}q), %\\
%\label{e2}
%\alpha \psi(q) &=&  e^{i <g_{{\rm o}} %\phi,\alpha> } \psi(q),
%\end{eqnarray}
%where $\; g_{{\rm o}} \in \mathcal G, \;$ $q %\in \mathcal G \phi,$ and
%$\alpha \in A$.
%Eqs. (\ref{e1}) and (\ref{e2})   define the IRS %of $B(2,1)$ induced from
%{\it each} $\; \phi \in
%A^{'} \;$
%and {\it each} irreducible representation $\; U %\;$ of
%$\;L_{\phi}.\;$
%The `Jacobian'
%$\frac{\rm{d}\mu_{g_{{\rm o}}}}{\rm{d}\mu} $
%of the group transformation is known as the \it %Radon$-$Nikodym \normalfont derivative of
%$\mu_{g_{{\rm o}}}$ with respect to $\mu$ and %ensures that the resulting IRS of $B(2,1)$ are %unitary.

%\noindent
% The central results of induced representation %theory
%(\cite{Wigner}, \cite{Mackey}, \cite{Mackey1}, %\cite{Simms},
%\cite{Isham} and references therein) are the %following:
%\begin{enumerate}
%\item {Given the topological restrictions on %$\; \mathcal B= A
%\rtimes_{T} \mathcal G \;$
%(separability and local compactness), any %representation of $\;
%\mathcal B, \;$  constructed by the method %above,  is irreducible
%if the representation $\; U \;$ of $\; L_{\phi} %\;$ on $\; D \;$
%is irreducible. Thus an irreducible %representation of $\;
%\mathcal B \;$ is obtained for {\it each} $\; %\phi \in
%%\hat
%A^{'} \;$
%and {\it each} irreducible representation $\; U %\;$ of
%$\;L_{\phi}.\;$} \item {If $\; \mathcal B= A
%\rtimes_{T} \mathcal G \;$ is a
%regular semidirect product (i.e., $\;
%%\hat
%A^{'} \;$ contains a Borel
%subset which meets each orbit in $\;
%%\hat
%A^{'} \;$ under $\; \mathcal
%H \;$ in just one point) then {\it all} of its %irreducible
%representations can be obtained in this way}.
%\end{enumerate}

%\noindent
%%Two remarks are in order regarding the %representations of
%%$B(2,1)$
%%obtained by the above
%%construction
%%\begin{enumerate}

%\item{
%\noindent
%As it is explained in \cite{Melas} \normalfont
%the subgroup $
%L^{2}(P_{1}(R),\lambda ,R)$ of
%$B(2,1)$
%%=
%%L^{2}(P_{1}(R),
%%%\newline
%%\lambda ,R)
%%\rtimes_{T}  G \;$
%is
%topologised as a (pre) Hilbert space by using a natural measure $\lambda$
%on $P_{1}(R)$ and by introducing
%a scalar product into $L^{2}(P_{1}(R),\lambda ,R).$
%If $ R^{4}$ is endowed with the natural metric topology then
%the group $  G=SL(2,R),$ considered as
%a subset of $ R^{4},$ inherits the induced topology on $G.$
%In the product topology of $
%L^{2}(P_{1}(R),\lambda ,R)
%\times   G,$ $B(2,1)$
%%_{c}
%%\;$
%is a non$-$locally compact
%group; the proof follows the corresponding proof
%%, without substantial change, Cantoni's
% \cite{Cantoni} for the original BMS group $B$.
%%, see also \cite{mac3}). (
%So far, the supertranslation space $F_{1}(N^{+})$ has been defined as a space of truly \it arbitrary \normalfont
%homogeneous  functions of degree one. This has been merely for clarity; for physical applications, it is necessary to give this space additional %structure.
%In fact the supertranslation subgroup
%$F_{1}(N^{+})$ can be given \cite{Melas} many additional
%structures, and as a result,
%$\;
%L^{2}(P_{1}(R),\lambda ,R),
%\;$
%and therefore
%the group $ B(2,1)$ can be employed with many different topologies.
%The Hilbert type topology employed here appears to describe
%\cite{Crampin2} quantum mechanical systems in asymptotically flat space$-$times.
%%\cite{Crampin2}.
%%).
%Since in the Hilbert type topology
%$  B(2,1)=
%L^{2}(P_{1}(R),\lambda ,R)
%\rtimes_{T}  G $ is not
%locally compact the theorems  dealing with the irreducibility of
%the representations obtained by the above construction no longer
%apply (see e.g. \cite{Mackey1}). However,
%it can be proved \cite{Melas} that the
%induced representations obtained above \it {are} \normalfont irreducible.
%The proof is similar to
%%follows very closely
%the one given in \cite{mac5} for the case of
%the original BMS group $\; B. \;$
%}


%\item{Here it is assumed that  $\;  B(2,1)
%\;$ is equipped with the Hilbert topology.
%It is of outmost significance that it can be proved %\cite{Melas}
%that in this topology $\;  B(2,1)
%\;$ is a regular semi$-$direct$-$product.
%The proof
%follows the corresponding proof \cite{Piard1,Piard2} for the %group $B$.
%Regularity amounts  to the fact \cite{Mackey} that
%${L^{2}}^{'}(P_{1}(R),\lambda ,R)$ can have no
%%(or cannot have)
%equivalent classes of quasi$-$invariant measures $\; \mu \;$ %such that the action of $\;
%\mathcal G \;$ is strictly ergodic with respect to $\; \mu. %\;$
%When such measures $\; \mu \;$ do exist it can be proved %\cite{Mackey}
%that
%an irreducible representation
%of the group, with the semi$-$direct$-$product structure at %hand,
%may be associated with each such measure $\mu$, that is not %equivalent to any of the IRS
%constructed by the Wigner$-$Mackey's inducing method.
%In a different topology it is not known if $\;  B(2,1)
%%_{c}
%\;$ is a regular or irregular semi$-$direct$-$product.
%Irregularity of $\;  B(2,1)$ in a topology different from the %Hilbert
%topology would imply that there are IRS of $\;  B(2,1)$ that %are not unitary equivalent to any of the
%IRS obtained
%%above
%by the inducing construction.
%Strictly ergodic
%actions are notoriously hard to deal with even in the locally
%compact case. Indeed, for locally compact non$-$regular %semidirect
%products, there is no known example for which all inequivalent
%irreducibles arising from strictly ergodic actions have been
%found. For the other 41 groups defined in
%\cite{mac1} regularity has only been proved for $B$ %\cite{Piard1,Piard2}
%when $B$ is equipped with the Hilbert topology.
%Similar remarks apply to all of them regarding IRS arising %from
%strictly ergodic actions in a given topology.
%}

%\end{enumerate}













\section{Extension of Wigner$-$Mackey's theory
%Orbits and ''little groups''
%''Little groups'' and inducing construction
}
\label{sec3}


In 1939 Wigner \cite{Wigner}
%broke new grounds in mathematics,
%by
gave the first explicit treatment of infinite dimensional representations of the Poincare group $P$, a \it ten$-$dimensional \normalfont Lie group. He constructed the IRS of $P$ by inducing
from the IRS of some of its subgroups, the ``little groups''. Mackey proved \cite{Mackey0,Mackey00}
that Wigner's inducing construction can be applied to any separable locally compact semidirect product
$\mathcal B = A  \rtimes_{T} \mathcal G $ specified by the continuous action $T: \mathcal G
\longrightarrow  {\rm Aut} (A).$
%The $T-$action of $\mathcal G$ on $A$ induces a dual
%$T^{\prime}-$action, given by equation (\ref{da}),
%of $\mathcal G$ on the topological dual $ A^{\prime}.$
Moreover Mackey proved \cite{Mackey0,Mackey00} that
when the semidirect product is regular,
%i.e., when
%there is a Borel set in $ A^{\prime}$ that meets each
%$\mathcal G-$orbit exactly once,
then \it all \normalfont
the IRS of $\mathcal B$ can be obtained by this inducing
construction.


It is noteworthy that both Wigner's work and Mackey's %monumental
work are grounded on a classical theorem of Frobenius
\cite{Frob}
according to which every ``imprimitive'' representation of
a finite group is induced in a certain canonical fashion
by a representation of one of its subgroups. The notion of
``imprimitive'' is not relevant here.

\subsection{Need for an extension}
\label{extension}
What is relevant here is that
Mackey initiates his study \cite{Mackey} by assuming that
$A$ is abelian, and that both
 the
abelian group $A$ and the group $\mathcal G$ in the semidirect product
$\mathcal B = A  \rtimes_{T} \mathcal G $ are locally compact.
In the applications to Physics we are considering here,
it is assumed that $A$ has additional
structure, it is also a vector space,  with
vector addition being the group multiplication.

It is well known that every Hausdorff topological vector space is locally compact if and only if is finite. This implies in particular
that a Hausdorff infinite dimensional topological vector space is never locally compact. Hausdorff is a desirable mathematical property which it is  highly praised by Physicists since it allows to discern the points of the topological space under consideration in an way ideal for them. The same happens here; it is
expected and desired both $A$ and its topological dual $ A^{\prime}$ to be  Hausdorff topological spaces.






















In all the  groups $
\mathcal B = A  \rtimes_{T} \mathcal G
$ we consider in the present paper,
the abelian group $A$ is also a
finite dimensional or
an infinite dimensional topological vector space,
which, in both cases, is Hausdorff in the whole class of topologies they come equipped with.
In particular, on every Hausdorff finite dimensional
topological vector space $A$ there is a unique
topological vector space structure, whereas,
%On the
%other hand
in the case of
%Hausdorff
infinite dimensional topological vector spaces $A$
 there is a wide range of allowed topologies,
%  class of allowed topologies is
  stated in Section
 \ref{top}. In all these topologies $A$ is
 Hausdorff.
We conclude, that for the groups
$
\mathcal B = A  \rtimes_{T} \mathcal G
$ we consider in the present paper,
Wigner$-$Mackey's theory applies, as it is,
when $A$ is a finite dimensional vector space.














The key new feature \cite{mac1} of the original BMS group $B$,
as well as of all possible analogues of $B$, both real and
in any signature, or complex, with all possible notions
of asymptotic flatness ``near infinity'', is the enlargement
of the finite dimensional vector space of translations
to the infinite dimensional vector space of supertranslations. This turns $B$, as well as of all possible analogues of $B$,
into infinite dimensional Lie groups.


It is precisely this new feature which calls for an extension of
Wigner$-$Mac$-$\newline key's theory. The required extension needs to
 give answers to the following problems, emanating
precisely from the infinite dimensions of the supertranslation
space, which are not addressed by Wigner$-$Mackey's theory.
\begin{enumerate}
\item{\it{First problem. The
``little groups'' problem:
Calculation of the ``little groups'' when the abelian subgroup $A$ of $\mathcal B = A \rtimes_{T} \mathcal G $   is also an  infinite dimensional vector space.} \normalfont \newline
In a nutshell Wigner$-$Mackey's inducing construction  allows to build up the
representations of the whole group out of
representations of some of its subgroups, the
``little groups.''
Wigner$-$Mackey's theory applies to
semidirect products $\mathcal B,$
such that, when $A$ is also a vector space,
it is a finite dimensional vector space.
When $A$ is
  a finite dimensional vector space then
 its topological dual $ A^{\prime}$ is also a finite
 dimensional vector space.







 \hspace{0.4cm} There is a natural
 bijection
\be
\label{bij}
\mathcal G / \mathcal G_{f} \longrightarrow \mathcal G f,
\ee
$g \mathcal G_{f} \longrightarrow g f,$  $f \in A^{\prime},$
between the coset space $\mathcal G / \mathcal G_{f}$ and the orbit $\mathcal G f$ of $\mathcal G$ in $ A^{\prime}.$ $G_{f}$ is the ``little group'' of $\mathcal G $ at the point $f.$ A complete classification of the $\mathcal G$ orbits
on $A^{\prime},$
when $A^{\prime}$  is a  finite dimensional
vector space, is a fairly easy task (see e.g. \cite{Barut}, p. 516). This led
Wigner and Mackey
%in their initial considerations
\cite{Wigner,Mackey0,Mackey} to
search for the ``little groups''  and
find them by classifying the $\mathcal G$ orbits
in $A^{\prime}.$ The situation changes drastically
when $A^{\prime}$ is an infinite dimensional vector space. Bijection (\ref{bij}) still holds,
 but a complete classification of the
$\mathcal G$ orbits in $A^{\prime}$ is not generally feasible. Consequently  the ``little groups'' cannot now be found by
classification of the
$\mathcal G$ orbits in $A^{\prime}$
and a different approach is required.
}

\vspace{0.2cm}
\item{\it{Second problem.
%\normalfont
The irreducibility problem:
%} \normalfont
%\it{
When $\mathcal B = A \rtimes_{T} \mathcal G $ is not locally compact it is not guaranteed
%we do not know
%if
that
the representations of $\mathcal B$ obtained by Wigner$-$Mackey's inducing construction are irreducible.}
\normalfont \newline
When $A$ is an infinite dimensional vector space
$\mathcal B$ is not locally compact, at least in the class of topologies $A,$ and subsequently
$\mathcal B,$ come equipped with
in the present paper; these topologies are elaborated in Section \ref{top}.
In Section \ref{top} we concentrate on the ``reasonable'' topologies for $B(2,1)$. However similar
topological structures are eligible
% comes equipped with but similar topological
for \it any \normalfont BMS group $\mathcal B$, and for \it any \normalfont of its analogues, in any number
$d$, $d\geq 3$, of space$-$time dimensions.
When $\mathcal B$ is not locally compact Mackey's theorems 14.1
and 14.2 in \cite{Mackey0},
which prove the irreducibility of the representations of $\mathcal B$ derived by the inducing construction,
do not no longer apply.



\vspace{0.2cm}
\item{\it{Third problem.
%\normalfont
The exhaustivity problem:
When the abelian group $A$
 is also an  infinite dimensional vector space, special care is needed in order to prove that the inducing construction is exhaustive.
} \normalfont
\normalfont \newline
When $\mathcal B$ \normalfont
is a regular  semidirect product
Wigner$-$Mackey's inducing construction
is exhaustive, i.e., \it all \normalfont
IRS of $\mathcal B$ are induced
from
%the
IRS of
%the
``little groups''.
%Wigner$-$Mackey's inducing construction
%is exhaustive, i.e., \it all \normalfont
%IRS of $\mathcal B$ are induced
%from
%%the
%IRS of
%%the
%``little groups'', when
%$\mathcal B$
%is a \it{regular} \normalfont semidirect product.
Regular semidirect product
$\mathcal B$ \normalfont means
that the action of $ \mathcal G$ on
$A^{\prime} $ is not pathological, and as a
result of this,
%that
the $ \mathcal G-$orbit$-$space $ A^{\prime} $
%this orbit space
is nice in a measure$-$theoretic
sense.



\hspace{0.4cm} When the space of $ \mathcal G-$orbits
in $ A^{\prime} $ is not nice, while
a given $ \mathcal G-$orbit and an irreducible
representation of the ''little group'' of a point
on that orbit still define an irreducible unitary
representation of $ \mathcal G,$ there will be still others not obtained by the inducing construction.




\hspace{0.4cm} A sufficient condition which
ensures the regularity of $\mathcal B$ (\cite{Mackey}, p.43, p.141),
is the existence of a
%Then there is
a Borel cross section for the
orbits in the action of $ \mathcal G$ on
$ A^{\prime},$ i.e., the existence of
%there is
a Borel set in $ A^{\prime} $ that meets each
$ \mathcal G-$orbit
exactly once.



\hspace{0.4cm}The real purpose of this condition
%hypothesis
about the existence of a Borel cross section is that it implies that
$ A^{\prime} $ can have no $ \mathcal G-$quasi$-$
invariant measures $\mu$  such that the action of
$ \mathcal G$ is strictly ergodic with respect to
$\mu.$ Whenever such measures $\mu$ do exist, an
irreducible representation of $\mathcal B$ may
be associated with each that is not equivalent to
any of the IRS of $\mathcal B$ induced from the IRS of
the ``little groups''. The nature of these representations is such that one can well despair
of ever obtaining a classification for them as
complete and satisfying as that available for
%\it{
regular
%} \normalfont
semidirect products.





\hspace{0.4cm}When the abelian group $A$
 is also a finite dimensional vector space,
then searching for a Borel cross section which
intersects each $ \mathcal G-$orbit in
$ A^{\prime} $ exactly once, is a fairly easy task
(see e.g. \cite{Simms}, p. 59).
However, when the abelian group $A$
 is also an  infinite dimensional vector space,
 as it is actually the case for both the original
 BMS group $B$ \cite{mac3} and its generalizations \cite{mac1}, and
the group $B(2,1)$ introduced here and its generalizations \cite{Mel8},
the
 situation is different.
 Then  $ A^{\prime} $ is also  an  infinite dimensional vector space.
We note that, as we clarified in Subsection \ref{extension}, at least in the class of topologies
elaborated in Section
 \ref{top}, when the abelian group $A$
 is also an  infinite dimensional vector space,
then both the BMS group $B$ in question and its
analogues, in any number $d$ of space$-$time dimensions,
$d \geq 3,$
%being \it any BMS group
%being
%the original
% BMS group $B$ \cite{mac3} and its generalizations %\cite{mac1}, and
%the group $B(2,1)$ introduced here and its generalizations %\cite{Mel8},
is not locally compact and vice versa.



 %as well.




\hspace{0.4cm} As a result, in this case, it is not generally feasible to classify completely \cite{mac4,Melas} the orbits
of the $\mathcal G$ action on $A^{\prime}$, let alone
% are too spacious
%to allow, in general, an easy classification of the
%%$\mathcal G$
%orbits of a group action
%on them.
%%$A^{\prime}.
%In fact when $A^{\prime}$ is an
%infinite dimensional Hilbert  space
%%Now, in general,
%the orbits of the $\mathcal G$ action
%on it, in general,
%%$\mathcal G$ orbits
%are parameterized by countably infinite sets of %parameters, and only a partial
%classification of them is feasible (see e.g. %\cite{mac4,Melas}).
to search for a a Borel set in $ A^{\prime} $ which meets each $ \mathcal G-$orbit
 in $ A^{\prime} $ exactly once. We conclude
 that when $A$
 is also an  infinite dimensional vector space,
 then the existence of a Borel cross section
 which meets each $ \mathcal G-$orbit in
  $ A^{\prime} $ exactly once, cannot be used
  as a sufficient condition, which can be
  practically implemented, for the regularity
  of the semidirect product $\mathcal B = A \rtimes_{T} \mathcal G, $ and new, different
 sufficient conditions which ensure
 the regularity
  of the semidirect product $\mathcal B = A \rtimes_{T} \mathcal G$  need to be introduced.





\hspace{0.4cm} One more word of caution is in order
here, word of caution brought to the fore again by the infinite dimensions of $A$, when $A$ is also an  infinite dimensional vector space.
The new sufficient conditions
which need to
be introduced in order to ensure the
regularity of $\mathcal B,$
%depend on the
involve the
topologies of $A$ and of $A^{\prime}$.
%with which $A^{\prime}$ comes equipped with.




\hspace{0.4cm}In a nutshell there is an interplay between the class of functions
in $A$, the class of functions in  $A^{\prime},$ and the topologies of
$A$ and $A^{\prime}:$ It is precisely the arbitrariness in the class of functions allowed in $ A,$ expounded in Section \ref{top},
 which permits a wide
range of choices of ``reasonable'' topologies for $ A$. The key point is that the type of functions in $ A$
determines the type of functions in
$A^{\prime}:$ Roughly speaking, the
``smoother''
the functions in $ A$ are, the ``rougher'' the generalized functions in $A^{\prime}$ turn out to be.  The functions in $A^{\prime}$ allow in turn
a variety
%wide range
of choices of ``reasonable'' topologies for $A^{\prime}$. To conclude:


\begin{enumerate}
\item{When the abelian group $A$ in
the semidirect product $\mathcal B = A \rtimes_{T} \mathcal G$ is also an infinite dimensional vector space new sufficient conditions are needed to insure the regularity of $\mathcal B.$
}
\item{The new sufficient conditions entail the topologies of $A$ and of its topological dual $A^{\prime}.$  }
\end{enumerate}
























} }


\end{enumerate}

%$B(2,1)$ is an infinite dimensional Lie group. %Infinite-dimensional ("Hausdorff" is part of the %definition) topological vector spaces are never %locally compact.


The  new feature  of the original BMS group $B$,
as well as of all possible analogues of $B$ \cite{mac1}, the enlargement
of the finite dimensional vector space of translations
to the infinite dimensional vector space of supertranslations, is also shared by $B(2,1)$ and its
analogues \cite{Mel8}. In fact it is a common characteristic
of \it any \normalfont BMS group $\mathcal B$, and its analogues, in any number
$d$, $d\geq 3$, of space$-$time dimensions.
 This turns $\mathcal B$, as well as of all possible analogues of $\mathcal B$, in any number
$d$, $d\geq 3$, of space$-$time dimensions,
into infinite dimensional Lie groups.
Thus the required extension, which has been reduced above to the solution of three problems, is needed in order to construct
the IRS of all these groups.


In Subsections \ref{little} and \ref{solutions}
we find solutions to the three problems posed by the
required extension of Wigner$-$Mackey's theory.
These solutions apply both to the original BMS group $B$
and also to $B(2,1).$ They also apply, possibly with
appropriate modifications in each particular case,
to \it any \normalfont BMS group $\mathcal B$, and to \it any \normalfont of its analogues, in any number
$d$, $d\geq 3$, of space$-$time dimensions.
In Subsection
%\ref{orbits} and
\ref{little}
we find a solution to the First problem
posed by the needed extension of
Wigner$-$Mackey's theory. This solution
applies to any topology $\mathcal B$ comes equipped with.
The solution to the Second problem, given in
Subsection \ref{solutions}, applies to the Hilbert
and to the nuclear topologies. Finally, the solution to the Third problem, given in
Subsection \ref{solutions}, applies to the Hilbert
 topology only. Thus solution to all three problems
 posed by  the
required extension of Wigner$-$Mackey's theory are
given in the Hilbert topology.



%In Subsection \ref{solutions}
%also solutions to the Second problem
%and to the Third problem
%posed by the needed extension
%of
%Wigner$-$Mackey's theory are given.


















\vspace{0.5cm}
%\noindent
In the next Subsection we clarify which is the Hilbert topology the group $B(2,1)$
comes equipped with, and we point out
that $B(2,1)$ is not locally compact in this topology.
%In Subsections
%\ref{orbits} and \ref{little}
%we find a solution to the First problem
%posed by the needed extension of
%Wigner$-$Mackey's theory.
%In Subsection \ref{solutions}
%also solutions to the Second problem
%and to the Third problem
%posed by the needed extension
%of
%Wigner$-$Mackey's theory are given.


\subsection{
%\bf{
$B(2,1)$ \ is \ not \ locally \ compact \ in \ the \ Hilbert \ topology
%}
}




%$\bf{B(2,1) \ is \ not \ locally \ compact \ in \ the \ Hilbert \ %topology }$
%\normalfont


\noindent
%As it is explained in \cite{Melas} \normalfont
%the subgroup $
%In this paper
$L^{2}(P_{1}(R),\lambda ,R)$
%of
%$B(2,1)$
%=
%L^{2}(P_{1}(R),
%%\newline
%\lambda ,R)
%\rtimes_{T}  G \;$
is
topologised \cite{Melas} as a (pre) Hilbert space by using a natural measure $\lambda$
on $P_{1}(R)$ and by introducing
scalar product (\ref{ip}) into $ L^{2}(P_{1}(R),\lambda ,R).$
The norm defined by this scalar product  induces a metric in
$ L^{2}(P_{1}(R),\lambda ,R). $
%The supertranslation space
$ L^{2}(P_{1}(R),\lambda ,R)$ is endowed
with the topology whose open sets are the
balls determined by this metric, the Hilbert
topology; the Hilbert topology is also called in the
literature metric, norm, or, strong topology.
%with the Hilbert topology; i.e. the topology
%whose open sets are the balls determined by this metric.
If $ R^{4}$ is endowed with the natural metric topology then
the group $  G=SL(2,R),$ considered as
a subset of $ R^{4},$ inherits the induced topology on $G.$














%In
We name the product topology of $
L^{2}(P_{1}(R),\lambda ,R)
\times   G,$ Hilbert topology,
to emphasize that $L^{2}(P_{1}(R),\lambda ,R)$
is endowed with the topology induced by the
scalar product (\ref{ip}) of the Hilbert space
$L^{2}(P_{1}(R),\lambda ,R).$
With the Hilbert topology, $B(2,1)$
becomes a topological group, in particular
%$B(2,1)$ Hilbert topology,
%to em
%_{c}
%\;$
%is
a non$-$locally compact
group; the proof follows the corresponding proof
%, without substantial change, Cantoni's
 \cite{Cantoni} for the original BMS group $B$.
%, see also \cite{mac3}). (
%So far, the supertranslation space $F_{1}(N^{+})$ has been defined %as a space of truly \it arbitrary \normalfont
%homogeneous  functions of degree one. This has been merely for %clarity; for physical applications, it is necessary to give this %space additional structure.
%In fact the supertranslation subgroup
%$F_{1}(N^{+})$ can be given \cite{Melas} many additional
%structures, and as a result,
%%$\;
%%L^{2}(P_{1}(R),\lambda ,R),
%%\;$
%%and therefore
%the group $ B(2,1)$ can be employed with many different topologies.
%The Hilbert type topology employed here appears to describe
%\cite{Crampin2} quantum mechanical systems in asymptotically flat %%space$-$times.
%%\cite{Crampin2}.
%%).





































































%$\mathcal G$ acts by matrix multiplication on
%$A$ from the left. Denote this by $\mathcal G \times A %\longrightarrow A$ with
%\be
%(g,\alpha)  \longmapsto g \cdot \alpha.
%\ee

\subsection{Orbits and ``little groups''}
\label{orbits}
Many symmetry groups  in Physics, which are isometry groups
of the underlying manifolds on which the dynamics unfolds,
 have the
structure of a semidirect product
\be
\label{sg}
\mathcal B = A  \rtimes_{T} \mathcal G,
\ee
where $A$ is a finite dimensional vector space, in particular $A=K^{d}$, $d\geq2$, $K$ is either the field $R$ of real numbers, or $C$  of complex numbers; thus
$A$ is the $K$$-$vector space of $d$$-$component column vectors with entries in $K$. $A$, being a vector space,
is an abelian group which is assumed to be a normal subgroup of $\mathcal B$. Now let $s: A \times A \longrightarrow K$ be any scalar product (i.e. any non$-$degenerate symmetric $K$$-$bilinerar form) for
$A$.
%, denoted
%\be
%s(\upsilon,w)=\upsilon \bullet w.
%\ee
$\mathcal G$ is the identity component of the ``Lorentz group''  of linear transformations preserving $s$.
%this scalar product.
The remaining structure of the semidirect
product is specified by the action $T$ of
$\mathcal G$ on $A$ given by
\be
T(g)\alpha= g \cdot \alpha,
\ee
where $g \in \mathcal G, \ \alpha \in A,$ and $g \cdot \alpha$ is matrix multiplication of $g \in \mathcal G$ from  the left with $\alpha \in A$. Examples of symmetry groups with the semidirect product structure (\ref{sg})
 are the connected Euclidean group and the connected Poincare group in any number of dimensions $d$, $d\geq2$, and also, the
connected Ultrahyperbolic group in any number of dimensions $d=2n$, $n=2,3... \ .$










In order to find the IRS of these groups, by applying
Wigner$-$Mackey's theory, the first step is
to find the ``little groups''.  Determination
of the ``little groups'' in turn typically requires (see e.g. \cite{Barut}, p. 510, 516, 517)  classification of the $\mathcal G$ orbits on the dual group $\hat{A}$ of $A$, or equivalently,
due to the bijection (\ref{bc}),
 classification of the $\mathcal G$ orbits on the topological dual  $A^{\prime}$ of $A$.
The key fact is that $A$ is a \it{finite} \normalfont dimensional vector space and
consequently its topological dual $A^{\prime}$ is a \it{finite} \normalfont dimensional vector space isomorphic to $A^{\prime}$,
and therefore, of
the same dimension as $A$. This makes
classification of the $\mathcal G$ orbits
on $A^{\prime}$ a fairly easy task (see e.g. \cite{Barut}, p. 516). In all cases the $\mathcal G$ orbits
are parameterized by  finite sets of parameters.










However, in the case of
%the BMS group
$B(2,1)$, $\mathcal G=G=
%{\rm
S
%}{\rm
L
%}
(2,R)$, and $A= {L^{2}}(P_{1}(R),\lambda ,R),$  which is an \it{infinite}
%$-$
\normalfont
dimensional Hilbert  space.
%From the
As we pointed out in Subsection \ref{sec2.3},
Reisz$-$Fr$\acute{e}$chet theorem for Hilbert spaces
%we know that
states that
the topological dual $A^{\prime}$ of $A$, is isomorphic to $A$.
Therefore, $A^{\prime}\simeq {L^{2}}(P_{1}(R),\lambda ,R)$, and
$A^{\prime}$ is itself an \it{infinite}
%$-$
\normalfont
dimensional Hilbert  space.
Hereafter we shall identify $A^{\prime}$ with
 ${L^{2}}(P_{1}(R),\lambda ,R)$ and we shall
 write $A^{\prime}= {L^{2}}(P_{1}(R),\lambda ,R),$
 with the understanding that $A^{\prime}$ and
 $L^{2}(P_{1}(R),\lambda ,R)$ are in fact isomorphic.






 Infinite dimensional Hilbert  spaces are too spacious
to allow, in general, an easy classification of the
%$\mathcal G$
orbits of a group action
on them.
%$A^{\prime}.
%In fact
When $A^{\prime}$ is an
infinite dimensional Hilbert  space
%Now, in general,
the orbits of the $\mathcal G$ action
on it, in general,
%$\mathcal G$ orbits
are parameterized by countably infinite sets of parameters, and only a partial
classification of them is feasible (see e.g. \cite{mac4,Melas}).
%In fact
This is
precisely what happens \cite{Melas} in the case of $B(2,1)$, classification of the $
%{\rm
S
%}{\rm
L
%}
(2,R)$ orbits on $A^{\prime}={L^{2}}(P_{1}(R),\lambda ,R)$ can only be partially
achieved.
%As a result, when $A^{\prime}$ is an infinite dimensional
%vector space,









To conclude, determining  the ``little groups'' by classifying
the $\mathcal G$ orbits is appropriate when $A$ is
a finite dimensional vector space; this
is precisely what Wigner and Mackey did in their
initial considerations \cite{Wigner,Mackey0,Mackey}.
On the other hand, when
$A$ is an infinite dimensional vector space,
and only, in general, a partial classification of the
$\mathcal G$ orbits can be attained, determining  the ``little groups'' by classifying
the $\mathcal G$ orbits is not appropriate,
and a different method should be followed. This is
precisely the problem which is addressed and solved
by the \it{Programme} \normalfont put forward in the
next Subsection.




\subsection{Determination of the ``little groups''
%Extension of Wigner$-$Mackey's theory
}
\label{little}
Hereafter, we restrict attention to semidirect products
$\mathcal B = A  \rtimes_{T} \mathcal G$
which share the new property of $B(2,1)$, namely,
their largest proper
%unique
normal abelian subgroup $A$, is also an
infinite dimensional vector space. In fact it is assumed that  $A$
is an infinite dimensional Hilbert space of real$-$valued
functions defined on a locally compact space $\mathcal Q$,
square integrable with respect to $\lambda$, $\lambda$ being a natural measure on $\mathcal Q$.







The subgroup $A$ may be topologized
as a (pre) Hilbert space by introducing a scalar product into $A$        and by using a natural measure $\lambda$ on $\mathcal Q$.
In the product topology of $A \times \mathcal G$, $\mathcal B$
becomes a topological group. In the case of $B(2,1)$, $A$ is topologized as a (pre)Hilbert space by introducing into $A$ the scalar product
\be
\label{ip}
<\alpha(\theta),\beta(\theta)>=\int_{P_{1}(R)}\alpha(\theta) \beta(\theta) {\rm d} \lambda(\theta)
\ee
where $\lambda$ is chosen to be the standard
normalized Lesbegue measure $ {\rm d} \lambda = \frac{{\rm d} \theta}{2 \pi}   $ on the circle.









As the  discussion in the previous Subsection suggests, in order to find the ``little groups'' when $A$ is an infinite dimensional vector space, we need to follow
another route \cite{mac6}. Firstly a definition is in order:

\begin{definition}
For each closed subgroup $\mathcal S$ of $\mathcal G$, define
its maximal invariant  vector subspace $\mathcal V$($\mathcal S$)
of $A^{\prime}$ by
\be
\mathcal V(\mathcal S)= \{ f \in A^{\prime}| \ s f = f \ \ \rm for \ \rm all \ s \in \mathcal S \}. \ee
\end{definition}
$\mathcal V$($\mathcal S)$ is closed since the $\mathcal G$ action on $A^{\prime}$
is continuous. In the case of $B(2,1)$, $A^{\prime}={L^{2}}(P_{1}(R),\lambda ,R),$
and consequently, $\mathcal V$($\mathcal S)$ being a closed subspace of
${L^{2}}(P_{1}(R),\lambda ,R),$ is itself a Hilbert space.
%This definition becomes more relevant
%$\mathcal V$($\mathcal S)$ is particularly useful when $\mathcal %S$ is a ``little group'' for $\mathcal B$. The following %definition particularizes the definition of the ``little group''
% given before.
%%in the previous section.
%\begin{definition}
%A subgroup $\mathcal S$ of $\mathcal G$ is a ``little group'' %for
%$\mathcal B$ if and only if
%\begin{enumerate}
%\item{$\exists \ f \in A^{\prime}, \ f \neq 0,$ such that $ s f = %f,$ for all $s \in \mathcal S.$}
%\item{If $\mathcal K$ contains $\mathcal S$ and leaves $f$ fixed, %$\mathcal K=\mathcal S$.}
%\end{enumerate}
%\end{definition}
%As a result, when $\mathcal S$ is a a ``little group'' for  %$\mathcal B$, then, for all $s \in \mathcal S$ and some %non$-$zero $f \in A^{\prime},$ $<s f, \alpha>=<f, \alpha>$ for %all $\alpha \in A,$
%where $<\cdot,\cdot>$ denotes the inner product in $A$. $\mathcal %S$ must be closed  since the $\mathcal G$ action on $A^{\prime}$ %is
%continuous and $f$ is a continuous linear functional.






Now the following holds \cite{mac6}:
\begin{theorem}
Every ``little group'' of $\mathcal B$ is a closed Lie subgroup of $\mathcal G$. The ``little groups'' fall into three classes:
\begin{enumerate}
\item{Discrete subgroups,}
\item{Connected Lie subgroups,}
\item{Non$-$connected  non$-$discrete Lie subgroups.}
\end{enumerate}
\end{theorem}
The identity subgroup $e \in \mathcal G$  belongs both to 1 and 2,
otherwise the three classes are non$-$overlapping.


When $A$ is infinite dimensional, the ``little groups'' of $\mathcal B$ may  be found, in principle, by following the \it{Programme} proposed by McCarthy
\cite{mac6}.

\vspace{1mm}


\noindent
\it{Programme} \normalfont

\begin{enumerate}
\item{Determine, up to conjugacy, all subgroups $
\mathcal S \subset \mathcal G$ belonging to the three classes 1, 2, and 3.}
\item{For each such $\mathcal S$ find the vector space $\mathcal V (\mathcal S)$.}
\item{Find the subgroups $\mathcal S$ for which $\mathcal V (\mathcal S)$ contains elements fixed under no properly larger group $\mathcal S^{'} \supset \mathcal S $. These last subgroups, and only these, are the ``little groups''.}
\end{enumerate}


\noindent
The following remarks are now in order:
%regarding this \it{Programme}. \normalfont
\begin{enumerate}
\item
{Regarding the first part of the \it{Programme} \normalfont we note that as it
%is
was
pointed out
in the previous Subsection, it is not appropriate, when $A$ is an infinite dimensional vector space,  to determine  the ``little groups'' by classifying
the $\mathcal G$ orbits. Instead, since
``little groups'' are subgroups of
$\mathcal G$, in order to find
the ``little groups'', we start by finding, up to
conjugacy, \it{all} \normalfont subgroups $\mathcal S$ of $\mathcal G$ belonging to the three classes 1, 2, and 3. Up to conjugacy, because conjugate ``little groups'' lead to equivalent IRS of of $\mathcal G.$ Finding the
subgroups $\mathcal S$ of $\mathcal G$ is a problem on its own right \cite{mac3}, which occasionally may be hard to solve \cite{Mel5},
or even does not have a complete solution \cite{mac6} with our
current knowledge of affairs.}



\item
{Regarding the second part of the \it{Programme} \normalfont we note that for \it{each} \normalfont subgroup $\mathcal S$ of $\mathcal G$, belonging to the three classes 1, 2, and 3, we search for non$-$zero
$f  \in A^{\prime}$ which are invariant under $\mathcal S.$ If there are such non$-$zero
$f  \in A^{\prime}$ which are invariant under  a subgroup $\mathcal S$ of
$\mathcal G,$  then the set $\mathcal V(\mathcal S)$ of
all such non$-$zero $f$, as we already pointed out, is a
vector space, in fact it is a Hilbert space, subspace of
$A^{\prime}={L^{2}}(P_{1}(R),\lambda ,R).$ It is appropriate now to draw attention to the following points:
\begin{enumerate}
\item {An element  $f$ is  non$-$zero as an element of the Hilbert space $A^{\prime}.$}

\item{It is easy to show that if $f_{1}$ and $f_{2}$
are two distinct elements of  $\mathcal V(\mathcal S)$ then the
orbits $\mathcal G f_{1}$ and $\mathcal G f_{2}$ are
also distinct.}

\item{If a subgroup $\mathcal S$ of $\mathcal G$ has non$-$zero invariant functionals $f  \in A^{\prime},$ then such a subgroup $\mathcal S$ \it does not \normalfont qualify yet as ``little group.'' Such a
    subgroup $\mathcal S$ is a \it potential \normalfont ''little group''.}
\end{enumerate}
%Those subgroups $\mathcal S$ of $\mathcal G$ which
%have non$-$zero invariant functionals $f  \in A^{\prime}$ %do not qualify yet as ``little groups''.
%They are \it{potential} \normalfont ``little groups''.



\hspace{0.4cm} $\mathcal S$ is a \it potential \normalfont ''little group''
because \it a priori \normalfont we cannot exclude the
possibility that \it all \normalfont the elements
$f  \in \mathcal V(\mathcal S)$  are also invariant under some bigger
group $\mathcal S^{*},$ where, $\mathcal S^{*} \subseteq \mathcal G.$ If this were the case then $\mathcal S$ would
not qualify as an actual ''little group''. On the other hand,
if for some group $\mathcal S,$ there is at least one
%only one
element
$f \in \mathcal V(\mathcal S),$ which is invariant only under  $\mathcal S,$ and which has no higher invariance, then the potential ''little group'' $\mathcal S $ does qualify as an actual ''little group''.}

\item {Regarding the third part of the \it{Programme} \normalfont we note that, as we explained before, in order to find the (actual) ''little groups'' we need to find those subgroups  $\mathcal S$ of $\mathcal G$ which have at least one
    %an
    invariant functional $f \in \mathcal V(\mathcal S)$
    with no higher invariance. This is a problem on its own right. Usually, after having identified $\mathcal V(\mathcal S),$  one proceeds with brutal force \cite{macMel,Mel6,Melas} and constructs explicitly such an $f$; a task which is by
    no means trivial and can take considerable amount of time and effort \cite{macMel,Mel6,Melas}. If such an $f \in \mathcal V(\mathcal S)$ is found then the potential ''little group''   $\mathcal S $ does qualify as an actual ''little group''.}
\end{enumerate}
%\vspace{-1cm}
%\noindent



The implementation of each part of the Programme is by no means trivial, and it poses its own problems and challenges \cite{mac3,mac4,mac6,Mel5,Mel6,Mel7,Melas}. In the case of $B(2,1)$ firstly we prove \cite{Melas} that the ``little groups'' are compact in the employed Hilbert topology. Knowing that the ``little groups'' are compact
facilitates subsequently the implementation of all parts of the Programme because
search now is restricted  to the compact subgroups $\mathcal S$ of $\mathcal G$ which belong to the three classes 1, 2, and 3. Moreover, the vector space $\mathcal V (\mathcal S)$ is calculated only for the compact subgroups $\mathcal S$ of $\mathcal G$, and only among these groups we isolate those which contain elements fixed under no properly larger
group.















%has been defined as a space of truly \it arbitrary \normalfont
%homogeneous  functions of degree one. This has been merely for clarity; %for physical applications, it is necessary to give this space additional %structure.
%In fact the supertranslation subgroup
%$F_{1}(N^{+})$ can be given \cite{Melas} many additional
%structures, and as a result,
%%$\;
%%L^{2}(P_{1}(R),\lambda ,R),
%%\;$
%%and therefore
%the group $ B(2,1)$ can be employed with many different topologies.
%The Hilbert type topology employed here appears to describe
%\cite{Crampin2} quantum mechanical systems in asymptotically flat %space$-$times.
%%\cite{Crampin2}.
%%).
%Since in the Hilbert type topology
%$  B(2,1)=
%L^{2}(P_{1}(R),\lambda ,R)
%\rtimes_{T}  G $ is not
%locally compact the theorems  dealing with the irreducibility of
%the representations obtained by the above construction no longer
%apply (see e.g. \cite{Mackey1}). However,
%it can be proved \cite{Melas} that the
%induced representations obtained above \it {are} \normalfont %irreducible.













%So far, the supertranslation space $F_{1}(N^{+})$ has been defined as a %space of truly \it arbitrary \normalfont
%homogeneous  functions of degree one. This has been merely for clarity; %for physical applications, it is necessary to give this space additional %structure.
%In fact the supertranslation subgroup
%$F_{1}(N^{+})$ can be given \cite{Melas} many additional
%structures, and as a result,
%%$\;
%%L^{2}(P_{1}(R),\lambda ,R),
%%\;$
%%and therefore
%the group $ B(2,1)$ can be employed with many different topologies.
%The Hilbert type topology employed here appears to describe
%\cite{Crampin2} quantum mechanical systems in asymptotically flat %space$-$times.
%%\cite{Crampin2}.
%%).
%Since in the Hilbert type topology
%$  B(2,1)=
%L^{2}(P_{1}(R),\lambda ,R)
%\rtimes_{T}  G $ is not
%locally compact the theorems  dealing with the irreducibility of
%the representations obtained by the above construction no longer
%apply (see e.g. \cite{Mackey1}). However,
%it can be proved \cite{Melas} that the
%induced representations obtained above \it {are} \normalfont %irreducible.







\subsection{
%Extension carried out
Extension of Wigner$-$Mackey's theory
accomplished in the Hilbert topology
%Solutions to the problems
}
\label{solutions}
As we pointed out in
Subsection \ref{extension}
Wigner$-$Mackey's theory
needs to be extended to be
applied to semidirect products
$\mathcal B = A \rtimes_{T} \mathcal G,$
when the Abelian group $A$
is also an infinite dimensional vector space. In order to be applied to this class of semidirect products Wigner$-$Mackey's theory needs to be extended in such a way
that solves the three problems stated in
Subsection \ref{extension}:
The ``little groups'' problem, the irreducibility problem, and the exhaustivity problem.



We note that Wigner$-$Mackey's theory is an enormously effective, systematic tool for
analyzing representations of many different groups,  has been used to good effect
by many workers, and has been extended in different directions over the years.
The extension given here is the appropriate  one in order to
study and construct the IRS of the BMS group $B$, and the IRS of its analogues,
in any number $d$
of space$-$time dimensions, $d\geq3$, and also in order to construct the IRS
of their supersymmetric counterparts.








In Subsection
%\ref{orbits} and
\ref{little}
we solved
%we found a solution to
the ``little groups'' problem.
In order to solve the problem
we restricted attention to semidirect products
$\mathcal B = A \rtimes_{T} \mathcal G,$
where $A={L^{2}}(\mathcal Q,\lambda ,R)$
is a separable infinite dimensional Hilbert space of real$-$valued
functions defined on a locally compact space $\mathcal Q$,
square integrable with respect to $\lambda$, $\lambda$ being a natural measure on $\mathcal Q$.





This is a large class of
semidirect products which encompasses both the original BMS
group $B$ \cite{mac3,mac8,mac4,mac5,mac6} and its generalizations \cite{mac1} and the group $B(2,1)$ %\cite{Melas}
and its generalizations \cite{Mel8}, when the functions which inhabit
$A$ are  square integrable with respect to the appropriate
measure in each case.




For this class of semidirect products,
when $A$ is endowed with the Hilbert topology, we can also
solve the other two problems: The irreducibility problem, and the exhaustivity problem.  It is noteworthy
that the   solution to the ``little groups'' problem,
given in Section \ref{little},
applies  also to all
``reasonable'' topologies
$A$ can be endowed with.
%comes equipped with.
These topologies
 are explicated in the next section.





  To conclude solutions to the
  three problems, stated in Section \ref{extension},
when
%$A$, and consequently $\mathcal B$ are employed with
$\mathcal B$ is employed with
the Hilbert topology,
are obtained as follows:
%1) Assure if they are real valued or complex valued
%2) Say that this is a wide range of semidirect products which
%encompasses the original B and its generalizations and the %B(2,1)
%and its generalizations in the Hilbert topology
%3) Say that  for this class of groups we have solutions for %the
%other two problems
%It is noteworthy
%that this   solution applies for all
%``reasonable'' topologies
%$A$ comes equipped with. These topologies
% are explicated in the next section.
%the solutions to the irreducibility and the exhaustivity %problems is as follows for completeness we include the little %groups problem
\begin{enumerate}
\vspace{0.2cm}
\item{\it{First problem.
The ``little groups'' problem: }
\normalfont
This problem was
%is
solved in
Subsection
%\ref{orbits} and
\ref{little}.
}



\vspace{0.2cm}
\item{\it{Second problem.
The irreducibility problem: }
\normalfont When $\mathcal B$ is not locally compact Mackey's theorems 14.1
and 14.2 in \cite{Mackey0},
which prove the irreducibility of the representations of $\mathcal B$ derived by the inducing construction,
do not no longer apply. However, when $\mathcal B$
is employed with the Hilbert or nuclear topology,
by using the results
of \cite{mac5} and  \cite{mac6} respectively,
we can prove that the representations of $\mathcal B$
obtained by Wigner$-$Mackey's inducing construction are
irreducible despite the fact that $\mathcal B$  is not
locally compact in the employed Hilbert or nuclear topology. In the case of $B(2,1)$, and
for the Hilbert topology,  we give the proof in \cite{Melas}.
}

%ερμου-κηφισιας-ηπειρου-κοτζια-συγγρου






\vspace{0.2cm}
\item{\it{Third problem.
The exhaustivity problem: \normalfont
As we pointed out in Subsection \ref{extension},
when $A$ is an infinite dimensional  vector space, and
hence
%when
$\mathcal B$ is not locally compact,
Mackey's classic regularity condition, i.e.,
the existence of a Borel cross section
 which meets each $ \mathcal G-$orbit in
  $ A^{\prime} $ exactly once, which
ensures the exhaustivity of Wigner$-$Mackey's
inducing construction,
cannot be used
  as a sufficient condition,  and
 new
 sufficient conditions
which insure
the exhaustivity of the IRS
of $\mathcal B$ constructed
by Wigner$-$Mackey's
``little group'' method
% the regularity
%  new, different
% sufficient conditions which ensure
% the regularity
%of the semidirect product $\mathcal B = A %\rtimes_{T} \mathcal G$
  need to be introduced.



\hspace{0.4cm}  This was done in
 two remarkable papers by Piard \cite{Piard1,Piard2} in the case where
 $\mathcal B$ is endowed with the Hilbert
 topology. Piard followed what Mackey did \cite{Mackey0,Mackey00},
when $A$ is a finite dimensional vector space, and he
appropriately adapted and modified it to the case
where $A$ is an infinite dimensional vector space.


\hspace{0.4cm} The sufficient conditions he
 introduced entail the strong topology
 of $A$ and the weak topology of $A^{\prime}$.
Piard  proved  \cite{Piard2} that the original
BMS group $B$ satisfies the sufficient conditions he introduced and so he proved
\it that in the case of $B$ Wigner$-$Mackey's
``little group'' method gives all the IRS of
$B.$ \normalfont This proves in particular
that McCarthy in \cite{mac3} and \cite{mac4} obtained all the IRS of $B$
and this renders his results even more
important.


\hspace{0.4cm} It is not a trivial task
to check if a BMS group $B$ in any number
of $d\geq3$ space$-$time dimensions
satisfies Piard's sufficient conditions.
We undertake this task in \cite{Melas} and
we prove that the group $B(2,1)$ satisfies
Piard's sufficient conditions and therefore
the IRS of $B(2,1)$ obtained here by
applying Wigner$-$Mackey's
``little group'' method are exhaustive.


\hspace{0.4cm}The first one
third of Piard's paper deals also with the case
where $A$ is endowed with the nuclear topology.
It is an interesting and important open  problem
to complete Piard's investigation, or apply
a totally different approach, and prove or
disprove that we obtain all IRS of $\mathcal B$,
by applying Wigner$-$Mackey's inducing construction, when $A$ is equipped with the
nuclear topology.


\hspace{0.4cm}This will inform us
in particular if McCarthy  in \cite{mac6}, where he
equipped $A$ with the nuclear topology, obtained all IRS of the original BMS group $B$, and moreover, if we obtain all IRS of $B(2,1)$
by applying Wigner$-$Mackey's inducing construction, if we endow $A$ with the
nuclear topology. Similar remarks apply to
any  any BMS group $B$ in any number
of $d\geq3$ space$-$time dimensions.


%   by Piard in an important work in the case %where B is equipped with the Hilbert topology  %who followed Mackeys' proofs suitablyadapted to %the case when A is an infinite dimesnional %vector space
}}
\end{enumerate}


%exhaustivity of the IRS constructed by inducing %cannot
%be invoked, and, as a result, there is no clear %idea,






%\subsubsection{$\bf{Irreducibility \ and \ %exhaustivity} $}

%Since in the Hilbert
%%type
%topology
%$  B(2,1)=
%L^{2}(P_{1}(R),\lambda ,R)
%\rtimes_{T}  G $ is not
%locally compact the theorems  dealing with the %irreducibility
%and exhaustivity of the IRS constructed by %Wigner$-$Mackey's
%theory
% %inducing construction
%%of
%%the representations obtained by the above %construction
%no longer
%apply (see e.g. \cite{Mackey1}).

%However,
%it can be proved \cite{Melas} that the
%IRS of $B(2,1)$
%constructed
%%in this paper,
%by using the extension of Wigner$-$Mackey's %theory
%expounded here,
%%obtained above
%\it {are} \normalfont irreducible. Moreover,
%in \cite{Melas}, by following Piard %\cite{Piard1,Piard2},
%we prove that in fact we obtain \it{all} %\normalfont
%IRS of $B(2,1)$ by applying Wigner$-$Mackey's %theory,
%despite the fact that $B(2,1)$ is not locally %compact
%in the employed Hilbert topology.


\vspace{0.4cm}
%\noindent


It is possible and desirable,
as it is explained in the next section,
to endow $B(2,1)$ with other topologies, different from the Hilbert
topology. It is not known if $B(2,1)$ is locally compact
in these other topologies.
%a topology different from the Hilbert topology.
Therefore,
if $B(2,1)$ is not locally compact in these other topologies
%in these new topologies,
%other topologies
%we do not know and
we should
examine anew
%, in these new topologies,
if
the representations of $B(2,1)$ we obtain with Wigner$-$Mackey's inducing construction are irreducible, and, if we obtain all of them.



\section{Other choices of topologies}
\label{top}

It is amusing that one can understand intuitively
\cite{mac8,Melas} that the ``little groups'' are compact in terms of
light intensity distributions on distant ``celestial'' spheres.
The dual action $T'$ of $ G$ on $ {L^{2}}^{'}(P_{1}(R),\lambda ,R)$
is given by equation (\ref{tzatziki}):
$$
(T'(g)f)(\sigma)=\kappa_{g}^{-2}(\sigma)f(\sigma g).
$$



This action is similar to the transformation law for
light intensity distributions on distant (``celestial'') 1$-$dim   spheres, i.e. circles, under $G$. The latter action appears with a $\kappa_{g}^{-1}$ rather than $\kappa_{g}^{-2}$ factor (see e.g. \cite{Greiner}, p.405, the discussion there refers to  4$-$dim space$-$time but it is easily
adapted to the case of 3$-$dim space$-$time); but the analogy works nevertheless. The $\kappa_{g}$ factor corresponds physically to a
blueshift factor. A subgroup $\mathcal S$ of $ G$ is a ''little group'' if it leaves a (non$-$zero) $f$ fixed under $T'.$  We showed
before that $ \mathcal S$ is closed. Consequently, if $ \mathcal S$
is non$-$compact, it must contain a sequence of pure boosts whose velocity parameter becomes arbitrary close to the speed of light.
For a light intensity distribution $f$ to be invariant under $\mathcal S$,  it must be such that, when subjected to arbitrary large
blueshifts over a region of the circle with arbitrary small complement
(the complement is redshifted), it remains the same. It is not surprising that there are no such non$-$zero $f's$, so that there cannot be any
non$-$compact ``little groups''.


In this paper we restrict attention to the case where
the supertranslation space $F(P(N^{+}))$
%$F_{1}(N^{+})$
is
%given additional
%structure
realized as
a separable Hilbert space, namely  $ L^{2}(P_{1}(R),\lambda ,R).$
We equip $ L^{2}(P_{1}(R),\lambda ,R)$ with the Hilbert
topology and topologize $B(2,1)$ with the product topology of
$L^{2}(P_{1}(R),\lambda ,R)\times   G.$
%The norm defined by the scalar product (\ref{ip}) induces a metric %in
%$ L^{2}(P_{1}(R),\lambda ,R).$
%%The supertranslation space
%$ L^{2}(P_{1}(R),\lambda ,R)$ is endowed
%with the Hilbert topology; i.e. the topology
%whose open sets are the balls determined by this metric.
There is a wide range  \cite{Crampin2} of
``reasonable'' topologies by which we can endow $B(2,1).$
This freedom we have in the choice of the topology for $B(2,1)$
stems
%arising
from the infinite$-$dimensional additive supertranslation
subgroup $F(P(N^{+}))$ of ``arbitrary'' real$-$valued functions
defined on the $P(N^{+}) \simeq S^{1}$.
%Also
The range of
choices available depends on the class of functions
allowed in $F(P(N^{+})).$


Regarding the class of functions allowed in $F(P(N^{+}))$
we note the following: In his original derivation of the
BMS group $B$ Sachs took supertranslations \cite{Sachs1}
to be twice continuously differentiable functions, since
they are coordinate transformations. Sach's original derivation of $B$ was superseded by that of Penrose \cite{Pen1,Pen2,Pen3},
who gave a precise and intrinsic derivation of $B$ as that group
of \it exact \normalfont conformal motions of the future
(or past) null boundary $\Im^{+}$ (or $\Im^{-}$) of
(conformally compactified weakly asymptotically simple)          space$-$times which preserve ``null angles''. Since truly arbitrary supertranslation functions describe symmetry transformations
in Penrose's sense, supertranslations need not have some minimum degree of smoothness.

$B(2,1)$ can also be derived in Penrose's sense, i.e., as as that group
of \it exact \normalfont conformal motions of the future
(or past) null boundary $\Im^{+}$ (or $\Im^{-}$) of 3$-$dim
(conformally compactified weakly asymptotically simple)          space$-$times which preserve ``null angles''. Consequently
similar remarks for the available freedom in the choice of
the degree of smoothness of the supertranslations apply also in
the case of $B(2,1).$







\indent
We pointed out before that
the dual action $T'$ of $ G$ on $ {L^{2}}^{'}(P_{1}(R),\lambda ,R)$
 is similar to the transformation law for light intensity distributions on distant (``celestial'') 1$-$dim   spheres under $G$, and this suggests, that there cannot exist any invariant,
 under non$-$compact  ``little group'' $\mathcal S$,
 square integrable $f \in {L^{2}}^{'}(P_{1}(R),\lambda ,R).$
However, the same reasoning gives rise to the possibility
%this suggests
that there may exist invariant
\it distributional \normalfont $f$ invariant under
non$-$compact  ``little groups''.

\indent
Indeed this is what happens in the 4$-$dim case \cite{mac6}, when $F(P(N^{+})),$ $P(N^{+}) \simeq S^{2}$ in the 4$-$dim case,
is chosen to be the space $C^{\infty}(S^{2})$ of infinitely differential functions on the 2$-$dim sphere. If a sufficiently fine (actually nuclear) topology is chosen for $C^{\infty}(S^{2})$ then non$-$compact  ``little groups'' arise associated with invariant distributions. We conjecture that the same is going to happen in the 3$-$dim case: If $F(P(N^{+}))$
is chosen to be the space $C^{\infty}(S^{1})$
of infinitely differential functions on the circle,
and equip this space with the nuclear topology then
non$-$compact  ``little groups'' are going to arise associated with invariant distributions.


\indent
IRS both in the Hilbert topology and in the nuclear topology induced by compact ``little groups'' describe bounded sources, whereas,  IRS in the nuclear topology induced by non$-$compact
``little groups''  describe scattering states \cite{mac6,Crampin2}. Another class of topologies for
$F(P(N^{+}))$ are the Sobolev topologies \cite{Crampin2}, which depend strongly on smoothness
properties (being defined by convergence in the mean of functions and derivatives). They are physically plausible in a gravitational  context because they  are adapted to initial$-$value problems for hyperbolic differential equations; in an ADM formulation of G.R. the evolution equations of the metric and the extrinsic curvature are hyperbolic equations.


%T(g)f=f
%T(g)F=F
%T(g1)f=F

%T(g1)f=T(g)F

%([T(g)]^{-1})T(g1)f=F

%([T(g)]^{-1})T(g1)f=F


%(g2f)(a)=f(g2^{-1}a)


%g1(g2f)(a)=g2f(g1^{-1}a)=f(g2^{-1}(g1^{-1}a))=f((g1g2)^{-1}a)






\section{Construction of the IRS of $B(2,1)$}

%\section{Results}
\label{sec4}
\subsection{Prerequisites}
\noindent
%Let $ A $ and $ \mathcal G $ be topological %groups, and let $ T $
%be a given homomorphism from $ \mathcal G $ %into the group of
%automorphisms ${\rm Aut} (A)$ of $A.$ %Suppose $A$
%is abelian and
Let
$\mathcal B = A  \rtimes_{T} \mathcal G $
%\bigcirc\!\!\;\!\!\!\;\!\!\!\!s \  _{T}
%\mathcal G $
be the semidirect product of $A$ and $ \mathcal G,$
specified by the continuous action $T: \mathcal G
\longrightarrow  {\rm Aut} (A).$
In section \ref{sec2} we pointed out that there is a
bijective correspondence between the dual group $\hat{A}$ of $A$ and the topological dual $ A^{\prime}$ of $ A$.
This bijection is also
useful here so we recall the explicit form of the
bijection between $\hat{A}$  and the  $ A^{\prime}$ and draw a conclusion not stated in Section \ref{sec2}.
 $\hat{A}$  and  $ A^{\prime}$ are isomorphic
as real vector spaces, and in particular as abelian groups. The isomorphism $j$ between  $\hat{A}$  and  $ A^{\prime}$ is given by equation
(\ref{isoj}):
$$
j:  A^{\prime}  \rightarrow  \hat{ A}, \ \ \
j(f)  =  e^{i f}. \nonumber
$$





Therefore every element $j(f)=\phi \in \hat{ A}$
is written
%uniquely
as $\phi=e^{i f}$, where
$f$ is determined uniquely by $\phi$
and vice versa. We have (equation (\ref{chi})):
%writing the character $\phi$ as
$$
%\label{chi}
\phi(\alpha)=e^{i f(\alpha)},
$$
$\alpha \in A.$
The action $T$ of $\mathcal G$ on $\hat{A}$
is defined by (equation (\ref{indact})):
%Indeed, any bijective map $b: A \longrightarrow A$,
%induces a map $\hat{A} \longrightarrow  \hat{A}$,
%$\phi  \longmapsto b \phi$, defined by
%\be
$$
%\label{indact}
(b \phi)(\alpha)=\phi(b^{-1}\alpha),
$$
%\ee
$b \in \mathcal G, \ \phi \in \hat{A}, \ \alpha \in A.$ The bijective correspondence
%$A \leftrightarrow  A^{\prime}$
$
%\label{bc}
\hat{ A} \leftrightarrow  A^{\prime}
$
is preserved pointwise by the $\mathcal G$ action
if and only if the action of $\mathcal G$
on $ A^{\prime}$ is defined by (equation (\ref{da})):
$$
%\label{da}
(gf)(\alpha)=f(g^{-1}\alpha),
$$
$g \in \mathcal G, \ f \in A^{\prime}, \ \alpha \in A.$





In Section \ref{sec2} we defined the ``little group''
$L_{\phi} $ of $ \phi,$ $\phi \in \hat{A}, $
to be the largest subgroup
of $\mathcal G $ which leaves $\phi $ fixed.
Similarly, we can define the ``little group''
$L_{f} $ of $ f,$ $f \in A^{\prime}, $
to be the largest subgroup
of $\mathcal G $ which leaves $f $ fixed.
It is an immediate consequence of equations
(\ref{chi}), (\ref{indact}), and  (\ref{da}),
that if $f$ is the unique element of  $A^{\prime}$
which corresponds to the element $\phi $ of
$ \hat{A}, $ then, $L_{f}=L_{\phi},$ i.e., we have:
\be
\label{iso}
\rm {If}  \ \ \it{f} \longleftrightarrow \phi \ \ \rm {then} \ \
\it L_{f}=L_{\phi}.
\ee




The last equality,  not mentioned in Section \ref{sec2},
is going to be useful in the subsequent construction of the IRS of $B(2,1)$.
One more remark is now in order before proceeding to the construction of the
IRS of $B(2,1)$.
As we  stated in Section \ref{sec2},
the set of characters which can be
reached from a character $ \phi $ by the
$  \mathcal G $ action on $\hat{A}$ (equation (\ref{indact}))   is called the orbit
of $ \phi $, and is denoted by $ \mathcal G \phi.$
There is a bijection between the orbit $ \mathcal G \phi$
and the coset space $\mathcal G / L_{\phi}$,
the set of left cosets of $L_{\phi}$ in $\mathcal G,$
(equation (\ref{omg}))
\be
\label{bij1}
 \mathcal G \phi \longleftrightarrow \mathcal G/L_{\phi}.
\ee





Similarly, the set of functionals which can be
reached from a functional $ f  $ by the
$  \mathcal G $ action on $A^{\prime}$ (equation ((\ref{da}))   is called the orbit
of $ f  $, and is denoted by $ \mathcal G f .$
There is a bijection between the orbit $ \mathcal G f $
and the coset space $\mathcal G / L_{f }$,
the set of left cosets of $L_{f }$ in $\mathcal G,$
%(equation (\ref{omg}))
\be
\label{bij2}
 \mathcal G f \longleftrightarrow \mathcal G/L_{f}.
\ee
Both bijections (\ref{bij1}) and (\ref{bij2})   will even be homeomorphisms \cite{Bour} (Appendix I), under quite general conditions.






From equations (\ref{iso}), (\ref{bij1}), and (\ref{bij2}), we conclude that if
$\phi$ is the unique element of  $\hat{A}$
which corresponds to the element $f $ of
$ A^{\prime}, $ then, there is a bijective correspondence between the orbits $\mathcal G \phi$ and
$\mathcal G f$, i.e.,  we have
\be
\label{bij3}
\mathcal G \phi  \longleftrightarrow \mathcal G f.
\ee
Therefore we can identify, if needed, the sets
$\mathcal G \phi$ and $G f$.
The previous comments also apply to $B(2,1)$, in which case we have,
 $A^{\prime} \simeq A= {L^{2}}(P_{1}(R),\lambda ,R)$, and,    $\mathcal G=G=
%{\rm
S
%}{\rm
L
%}
(2,R)$.




%From equations  we
%For a  given character
%$ \phi \in \hat{A},$ the largest subgroup $L_{\phi} $
%of $\mathcal G $ which leaves $\phi $ fixed is called
%the ''little group'' of $ \phi     $.
%In the case of $B(2,1)$








%action of G on A and A prime and A hat.


%The bijective correspondence
%$A \leftrightarrow  A^{\prime}$
%\be
%\label{bc}
%\hat{ A} \leftrightarrow  A^{\prime}
%\ee
%is preserved pointwise by the $\mathcal G$ action
%if and only if the action of $\mathcal G$
%on $ A^{\prime}$ is defined by
%\be
%\label{da}
%(gf)(\alpha)=f(g^{-1}\alpha).
%\ee



%We conclude that the ''Little group'' L$\phi$ is the same %with the
%''Little group'' L$f$In the case of $B(2,1)$A is this and
%A prime is this





%Writing the character $\phi$ as
%\be
%\label{chi}
%\phi(\alpha)=e^{i f(\alpha)},
%\ee
%one easily finds \cite{Melas}
%that the real$-$valued function $f$
%\begin{enumerate}
%\item Is continuous,
%\item Is additive, i.e., %$f(\alpha_{1}+\alpha_{2})=f(\alpha_{1})+f(\alpha_{2})$, %and,
%\item $f(\kappa  \alpha) = \kappa f(\alpha), \ \kappa %\in R.$
%\end{enumerate}
%Therefore
%$f$
%is a continuous linear functional on $ A$, that is, %$f$ belongs to the topological dual $ A^{\prime}$ of %$ A$.
%The map
%\begin{eqnarray}
%j:  A^{\prime} & \rightarrow & \hat{ A}, \ \rm{given %\ by} \nonumber \\
%\label{isoj}
%j(f) & = & e^{i f} \nonumber
%\end{eqnarray}
%provides an isomorphism between $\hat{ A}$ and $ %A^{\prime}$, as real vector spaces, and in particular %as abelian groups.
%We shall identify, if required, the sets
%$\hat{ A}$ and $ A^{\prime}$.
%The bijective correspondence
%$A \leftrightarrow  A^{\prime}$
%\be
%\label{bc}
%\hat{ A} \leftrightarrow  A^{\prime}
%\ee
%is preserved pointwise by the $\mathcal G$ action
%if and only if the action of $\mathcal G$
%on $ A^{\prime}$ is defined by
%\be
%\label{da}
%(gf)(\alpha)=f(g^{-1}\alpha).
%\ee















\subsection{Necessary data for the construction of the IRS}

To find explicitly the operators of the induced
representations of $B(2,1)$, it suffices to provide the information
stated in paragraphs 1 and 2 at the end of  Section \ref{sec2} for each of the orbit types.
It is convenient, in order to construct the operators of the IRS of
$B(2,1)$, to use the bijective correspondence between the
 dual group $\hat{A}$ of $A$ and the topological dual $ A^{\prime}$ of $ A$
 and the consequences of this correspondence, i.e., to use equations
(\ref{isoj}), (\ref{iso}), and (\ref{bij3}),  and restate the
% necessary
 information
% in order to construct the IRS of
%$B(2,1)$,
stated in paragraphs 1 and 2 at the end of  Section \ref{sec2} as follows:
%It follows from the previous discussion that
In order to give
the operators of the IRS of $B(2,1)$  explicitly
%%induced representations explicitly
it is
necessary to give the following information
\begin{enumerate}
\item{
An irreducible unitary representation $U$ of $L_{f}$ on a
Hilbert space $D$ for each $L_{f}.$}
\item{
A $ \mathcal G$$-$quasi$-$invariant measure $ \mu $
on each orbit $ \mathcal G f
\approx \mathcal G / L_{f};$
where $L_{f} $ denotes the ''little group'' of the base point
$ f \in A^{\prime} $ of the orbit
$ \mathcal G f.$
}
\end{enumerate}





Thus for \it each \normalfont orbit
$ G f \in A^{\prime}$, with
base point
$f,$ and \it each \normalfont irreducible unitary representation $U$ of $L_{f}$, we have
an irreducible representation of
$B(2,1),$ given by equation
(\ref{ukygvkhvkyutfkyufvmhgv}), which is
unitary, provided the orbit $ G f$ comes equipped with a measure which
is $ G$$-$quasi$-$invariant.
We conclude that in order to construct
the operators of the IRS of $B(2,1)$
we need to give the following data
in the prescribed order:
\begin{enumerate}
\item{``Little groups'' and their invariant functionals.}
\item{IRS of the ``little groups'' on a Hilbert space $D$.}
\item{A $ G$$-$quasi$-$invariant measure
on each orbit $  G f$.
%, $G=SL(2,R)$.
}
\end{enumerate}
We proceed now to carry out this task.




%We note that
%all the ``little groups'' are abelian. All IRS %of an abelian group are one$-$dimensional.
%\newline

\subsection{``Little groups'' and their invariant functionals}

\noindent
The calculation of the ``little groups'' goes
hand in hand with the calculation of their
invariant functionals, and it is achieved
by following the \it{Programme} \normalfont
stated in Subsection \ref{little}.




It is an important result
%\cite{Melas}
%it is proved
that when $B(2,1)$ is employed with the Hilbert topology \it {all} \normalfont ``little groups'' of $ B(2,1)$ are compact.
A formal proof is given in \cite{Melas}.
A heuristic, intuitive proof was given in Section
\ref{top}.

The maximal compact subgroup of $G$, where $G=SL(2,R)$, is
just the subgroup $SO(2).$ That is to say, if $K$ is a compact
subgroup of $G,$ then some conjugate $g K g^{-1}$ of $K$ is
a compact subgroup of $SO(2).$ In the representation theory
of $B(2,1)$, the little groups are only significant up to conjugacy. So, after a possible conjugation, we may take
every little group to be  a compact (or, equivalently, closed)
subgroup of $SO(2).$

%In particul
This makes the Theorem which follows more intelligible.
%The following Theorem,
This Theorem, proved in \cite{Melas},
describes in detail the  ``little groups'' of $ B(2,1).$
%the following Theorem is proved:
\begin{theorem}
\label{Lit}
The ``little groups'' $\;L_{f}\;$ for $B(2,1)$ are precisely the closed subgroups of $
%K=
S O (2)$ which contain the element $-I$, $I$ being the identity element of $SL(2,R)$. These are (A) $SO(2)$ itself, and (B)  the cyclic groups $C_{n}$ of
even order $n$.
\end{theorem}

The proof of the preceding Theorem involves identifying
$SO(2)$, and $C_{n},$ with $n$ even, as potential
``little groups'' and then proving that they do in fact
occur as actual ``little groups''. It is in this second part
of the proof that we proceed with brutal force
and actually construct in each case an invariant
functional with no higher invariance. This
construction, as we pointed out in Subsection
\ref{little}, is by no means a trivial task,
and can take in general \cite{macMel,Mel6}, and in fact it does so here,
%here in particular,
considerable amount of time and effort,
at least as much as required to identify the
potential ``little groups'' at the first place.

%of the proof which in general, as we pointed out in
%Subsection
% and here in particular, proceeds with brutal force
% by actually constructing an invariant functional












%\noindent
As we pointed out in Section \ref{sec3}, for a given ``little group'' $\;L_{f}\;$ the elements $f \in A^{'}$ which are invariant under $\;L_{f}\;$, i.e., the
elements $f \in A^{'}$ which satisfy
\begin{equation}
(T'(g)f)(\theta)= f(\theta),
\end{equation}
$\theta \in P_{1}(R),$
form a (Hilbert) subspace of ${L^{2}}(P_{1}(R),\lambda ,R)$. We denote this subspace by $L^{2}(L_{f})$.
We have the following Theorem \cite{Melas}:
\begin{theorem}
The Hilbert space $L^{2}(S O (2))$ of invariant vectors $f \in A^{'}$ under $S O (2)$ is:
\begin{equation}
L^{2}(S O (2))=\left \{ f \in L^{2}(P_{1}(R),\lambda ,R) \; | \;  f(\theta)=c, \; c \in R \right \}.
\end{equation}
So $L^{2}(S O (2))$ is just the one$-$dimensional space of constant real$-$valued functions defined on $P_{1}(R)$.
The Hilbert space $L^{2}(C_{n})$ of invariant vectors $f \in A^{'}$ under $C_{n}$ is:
\begin{equation}
L^{2}(C_{n})=L^{2}(E_{n}),
\end{equation}
where $L^{2}(E_{n})$ is the Hilbert space of square integrable real$-$valued functions defined on
\begin{equation}
E_{n}= \left \{ \theta \in  P_{1}(R)  \; | \;    0   <  \theta  < \frac{4 \pi}{n} \right \}.
\end{equation}
\end{theorem}
It is important to bear in mind that, as we emphasized
in Section \ref{sec3},  if $f_{1}$ and $f_{2}$ are two
distinct elements of $L^{2}(S O (2))$, or of $L^{2}(C_{n})$,
then the orbits $ G f_{1}$ and $ G f_{2}$,
%$G=SL(2,R)$,
are also distinct.


\it{Thus each $f \in L^{2}(S O (2)),$
and, each $f \in L^{2}(C_{n}),$ gives rise to a different
orbit $G f,$  and on each such orbit a continuous irreducible
unitary representation of $B(2,1)$ can be built in the
way summarized in paragraphs 1 and 2 of the previous Subsection. Representations built on distinct orbits are also
distinct, i.e., not unitary equivalent. }
\normalfont
%Therefore, on \it{each} \normalfont
%orbit
%\noindent



Moreover, in  \cite{Melas} it is shown that the Wigner$-$Mackey's inducing construction is exhaustive
despite the fact that $ B(2,1)$  is not
locally compact in the employed Hilbert topology.
This result is rather important because other group theoretical approaches
to quantum gravity which invoke Wigner$-$Mackey's inducing construction (see e.g. \cite{IshKak0,IshKak}) are typically plagued
by the non$-$exhau$-$ \newline stiveness of the inducing construction which results precisely from the fact that
the group in question  is not locally compact in the prescribed topology.
Exhaustiveness is not just a mathematical nicety:
If the inducing construction is not exhaustive one cannot
%simply
know if the most interesting information or part of it
is coded in the irreducibles which cannot be found by the Wigner$-$Mackey's inducing procedure.
These results, i.e.
compactness of the ``little groups'' and exhaustiveness of the inducing construction, not only are
 significant  for the group theoretical approach to quantum gravity advocated here, but also  they have
 repercussions \cite{Melas}  for  other approaches to quantum gravity.
%\newline


\subsection{IRS of the ``little groups'' on a Hilbert space $D$.}
%\section{Construction of the IRS of $B(2,1)$}
\noindent
%To find explicitly the operators of the induced
%representations of $B(2,1)$, it suffices to provide the %information
%cited in (i) and (ii) in section \ref{rep} for each of the %orbit types. We note that
%All the ``little groups'' are abelian.
The ``little groups'' of $B(2,1)$ are given in Theorem \ref{Lit} and are all abelian.
All the irreducible
representations of an
%All IRS of
 abelian group are one$-$dimensional. We have in particular:
\newline

\noindent
%\bf (i), \normalfont \hspace{0.2cm} \bf   (i$_{\alpha}$)
\bf (1).
\normalfont     \hspace{0.3cm} $L_{f}=SO(2)$.
\newline
\noindent
The IRS $U$ of $SO(2)$ are parameterized by an integer $\nu$ which for distinct representations takes  the values $\nu =...,-2,-1,0,1,2,... \ .$
Denoting these representations by $U^{(\nu)}$,
they are given by multiplication in one complex dimension
$D \approx C $ by
\begin{equation}
U^{(\nu)}(R(\theta))=e^{i \nu \theta},
\end{equation}
where $ R(\theta)=  \begin{pmatrix}
    cos\theta & -sin\theta \\
    sin\theta & cos\theta
  \end{pmatrix} \in SO(2)$.

\noindent
%\bf (i), \normalfont \hspace{0.2cm}
%\bf   (i$_{\beta}$)
\bf (2).
\normalfont     \hspace{0.3cm}
$L_{f}=C_{n}$, $n$ is even.
\newline
\noindent
The IRS $U$ of $C_{n}$ are parameterized by an integer $\lambda$ which for distinct representations takes  the values $\lambda =0,1,2,...,n-1.$
Denoting these representations by $U^{(\lambda)}$,
they are given by multiplication in one complex dimension
$D \approx C $ by
\begin{equation}
U^{(\lambda)} \left (R \left ( \frac{2 \pi}{n}j \right ) \right )=e^{i  \frac{2 \pi}{n} \lambda j},
\end{equation}
where $ j$  parameterizes the elements of the group $C_{n}$.


\subsection{ A $ G$$-$quasi$-$invariant measure on each orbit $Gf$.}
\label{invmeas}



%\noindent
%\bf (ii) \normalfont A $\; \mathcal G\;$$-$quasi$-$invariant
%measure $\; \mu \;$ on each orbit $\; \mathcal G \phi \approx
%\mathcal G / L_{\phi}\;$ is required, however, a $\; \mathcal G\;$$-$invariant
%measure $\; \mu \;$ on each orbit $\; \mathcal G \phi \approx
%\mathcal G / L_{\phi} \;$ will be given in all cases. We note that when $\mu$ is $\; \mathcal G\;$$-$invariant then $\frac{\rm{d}\mu_{g_{{\rm %o}}}}{\rm{d}\mu}(q)=1$, and this is precisely what happens in the case of $B(2,1)$. Moreover,  in the case of $B(2,1)$,
%$\mathcal G=G=$SL(2,$R$), and the ``little groups''
%$L_{\phi}$ are given in Theorem \ref{Lit}.
%\newline







\noindent
%\bf (ii) \normalfont
 A $\;  G\;$$-$quasi$-$invariant
measure $\; \mu \;$ on each orbit $\;  G f \approx
 G / L_{f}\;$ is required. However, a $\;  G\;$$-$invariant
measure $\; \mu \;$ on each orbit $\;  G f \approx
 G / L_{f} \;$
%will
can
be given in all cases.
%We note that when $\mu$ is $\; \mathcal G\;$$-$invariant then $\frac{\rm{d}\mu_{g_{{\rm o}}}}{\rm{d}\mu}(q)=1$, and this is precisely what %happens in the case of $B(2,1)$. Moreover,  in the case of $B(2,1)$,
%$\mathcal G=G=$SL(2,$R$), and the ``little groups''
%$L_{\phi}$ are given in Theorem \ref{Lit}.


\subsubsection{Invariance and quasi$-$invariance}

\noindent
It is useful at this point to recall some basic facts
from elementary measure theory \cite{Bourbaki}.
A measure $\mu$ on a measurable space $\mathcal M$ is called
 $\mathcal G$$-$invariant, or invariant with respect to
$\mathcal G$, if for any measurable set $E \subset \mathcal M$,
and $k \in \mathcal G$, the transformed measure $k \mu$, also
denoted by $\mu_{k}$, defined by
\be
\mu_{k}(E)=\mu(k^{-1}E),
\ee
%for any measurable set $E \subset \mathcal M$,
satisfies the relation
\be
\mu_{k}(E)=\mu(E),
\ee
for any measurable set $E \subset \mathcal M$.

Therefore a measure $\mu$ on a measurable space $\mathcal M$ is called
 $\mathcal G$$-$invariant, if
 \be
 \label{inv}
\mu(k^{-1}E)=\mu(E),
 \ee
for any $k \in \mathcal G$ and any $E \subset \mathcal M.$

The weaker condition of $\mathcal G$$-$quasi$-$invariant measure $\mu$, or measure
$\mu$ quasi$-$ \newline invariant with respect to $\mathcal G$, states that if $E$ has positive measure, then its image $k^{-1} E$ has also positive measure for any $k \in \mathcal G.$ Thus
\be
\label{con1}
\mu(E)>0 \Rightarrow \mu_{k}(E)>0,
\ee
for any $k \in \mathcal G.$
The last condition is equivalent to the statement that the measures $\mu$ and
$\mu_{k}$ have the same sets of measure zero, i.e.,
\be
\label{con2}
\mu(E)=0 \Rightarrow \mu_{k}(E)=0.
\ee



Quasi$-$invariance is necessary and sufficient for the existence of the Radon$-$\newline Nikodym (R$-$N) derivative
$\frac{\rm{d}\mu_{k}}{\rm{d}\mu}(p)$, $p \in \mathcal M$,
for all group elements $k \in \mathcal G.$ In the case
under consideration $\mathcal G= G = SL(2,R)$, and
$\mathcal M=G/L_{f}.$ Note that it is precisely the
quasi$-$invariance of $\mu$ with respect to $G$ that
allows the square root of the R$-$N derivative to occur as
a factor in (\ref{ukygvkhvkyutfkyufvmhgv}), and this is exactly what is needed to make
the representation of $\mathcal B$ in $H_{\mu}$,
given in (\ref{ukygvkhvkyutfkyufvmhgv}), unitary.

Construction of $\mathcal G$$-$quasi$-$invariant measures
on the
% orbits  classification of the
$\mathcal G$ orbits on the topological dual  $A^{\prime}$ of $A$  is a major mathematical problem in every group theoretical approach to quantum gravity which involves the
construction of the IRS of a symmetry group of the form $\mathcal B = A  \rtimes_{T} \mathcal G. $
In every approach this measure carries  important information.

When $\mathcal G$ is a not locally compact group
the construction of $\mathcal G$$-$quasi$-$invariant measures
on the
% orbits  classification of the
$\mathcal G$ orbits on the topological dual  $A^{\prime}$ of $A$ is an in general unsolved problem. This is the case
in Isham's approach (in Section \ref{other} we adopted a different notation; we denoted $\mathcal B$ by $\mathcal C$,
 $A$  by $\mathcal S$, and, $\mathcal G$ by $G$).
 In fact Isham conjectured that there are not any
$\mathcal G$$-$quasi$-$invariant measures
on the $\mathcal G$ orbits on the topological dual  $A^{\prime}$ in his approach and that it is likely that
such measures will  of necessity be concentrated
on genuine distributional elements in $A^{\prime}$.
If this is true, Isham concluded,
``the group representation theory
determines by itself precisely what is to be understood by `distributional geometry'  ''.
No matter how significant and insightful that is, it is still a conjecture.


The key fact in our approach which comes to our rescue is that
$\mathcal G=G=SL(2,R).$ $SL(2,R)$ is locally compact and therefore has a Haar measure. This is true for any
%The extended theory is necessary
%generalizations
%in order to construct the IRS
%of both
BMS group
$B=A \rtimes_{T} \mathcal G$, and for any of its analogues, in any number $d\geq3$
of space$-$time dimensions. Thus in all cases $\mathcal G$  is
locally compact.



If this had not been the case
we would have added as the fourth problem,
in Subsection \ref{extension}, the construction of
 $\mathcal G$$-$quasi$-$invariant measures
on the
% orbits  classification of the
$\mathcal G$ orbits on the topological dual  $A^{\prime}$,
to the three problems stated there, namely,
 the ``little groups'' problem, the irreducibility problem, and the
exhaustivity problem,
the solutions to which are necessary in order to
carry out the necessary extension of Wigner$-$Mackey's theory.


In our approach the  $\mathcal G$$-$quasi$-$invariant measures
on the
% orbits  classification of the
$\mathcal G$ orbits on the topological dual  $A^{\prime}$
carries the information about how the outcomes of physical
measurements distribute.





\subsubsection{The Haar measure}
The Haar measure is a generalization of the Lebesgue
measure to the case where the measurable space
$\mathcal M$ is a locally compact group $G$. The Haar measure
shares the key property of the Lebesgue
measure, namely it is invariant under translations.
Since the measurable space is a group it is natural to
distinguish left$-$invariance from right$-$invariance
since the translation is realized now by group multiplication.
In particular the following Theorem \cite{Haar} holds:
\begin{theorem}
Let $G$ be a locally compact group. Then there exists a unique, up to a positive multiplicative
constant, left$-$invariant measure, the Haar measure, on $G$. Similarly if
``left$-$invariant'' is replaced by``right$-$invariant''.
\end{theorem}



Let $ \Omega$ be the class of Borel sets in $G.$
A left invariant (right invariant) Haar measure on $G$
is a measure $\mu$ such that
\be
\label{la}
\mu(g^{-1}A)=\mu(A)
 \ee
($\mu(A g^{-1})=\mu(A)$), for all $g \in G$ and $A \in  \Omega.$ Note that according to the definition of
invariant measures given in equation (\ref{inv}) a left
invariant measure on $G$, is $G$$-$invariant.





Given a left Haar measure, one immediately obtains a right Haar measure, and
vice versa. In particular, if $\mu$  is a left (right) invariant
measure on $G$ then $\mu^{\prime}(A)=\mu(A^{-1})$
is a right (left) measure on $G$, where $A^{-1}$
contains the inverse elements of $A$.












%In the case of $B(2,1)$ the key fact is that $ G= SL(2,R)$ is %locally compact and
%hence we can apply the usual method of projecting down the %Haar measure from $G$
%to the $G$$-$orbits $G/L_{f}.$ To this end it is useful at %this point to recall
%the following Theorem from basic measure theory \cite{Haar}  %on locally compact groups:


In general, a left$-$invariant Haar measure on a locally compact group $G$, is not
right$-$invariant. When on a group $G$ the Haar measure is both left and right invariant
the group $G$ is called unimodular.

A subgroup of a locally compact group is locally compact. Therefore since $SL(2,R)$ is locally compact, its subgroups
%the groups
$SO(2)$, and
$C_{n}$, where $n$ is even, are locally compact. As a matter of fact,  the groups $SO(2)$, and
$C_{n}$, where $n$ is even, are compact, and hence closed,
 subgroups of $SL(2,R)$. Compact groups are unimodular.
Therefore the groups
$SO(2)$, and
$C_{n}$, where $n$ is even, are unimodular.
Moreover, the group $SL(2,R)$ is also
%  $SO(2)$, and
%$C_{n}$, with $n$ is even, are
unimodular \cite{Lang}.

The Haar measure on $SL(2,R)$ is given in \cite{Gel}, p. 214$-$215, the
Haar measure on $SO(2)$ is the usual
Lebesgue measure on $SO(2)$, and the Haar measure on $C_{n}$, with $n$ even, is the
counting measure on $C_{n}$ defined by:
\be
\mu: \ \mathcal P(C_{n}) \longrightarrow R_{\geq}, \quad \mu(E)=| E |,
\ee
where  $\mathcal P(C_{n})$ is the power set of $C_{n},$
$R_{\geq}$ is the set of non$-$negative real numbers, and $| E |$,
is the order of the set $E.$


\subsubsection{Projection of the Haar measure}


Consider the canonical map
\be
pr: \ G \longrightarrow G/L_{f},
\ee
where $G=SL(2,R)$, and $L_{f}=SO(2)$, or $L_{f}=C_{n}$, where $n$ is even.
It is a beautiful, intuitively  obvious, Theorem (\cite{Lang}, p. 37, Theorem 1)
that when $G$ and $L_{f}$ are unimodular, and $L_{f}$
is a closed subgroup of $G$,
the unique, up to an arbitrary multiplicative positive constant,
Haar measure on $G$  can be projected down uniquely, up to an arbitrary  multiplicative positive constant,
to a $G$$-$invariant  Haar measure on the coset space $G/L_{f}$, and hence
to a $G$$-$invariant  Haar measure on the orbit  $Gf.$


The Theorem asserts in particular that if $\mu_{G}$ and $\mu_{L_{f}}$
are the unique, up to an arbitrary multiplicative positive constant,
Haar measures on the groups $G$ and $L_{f}$ respectively, then there is
a unique, up to an arbitrary multiplicative positive constant,
  $G$$-$invariant  Haar measure $\mu_{G/L_{f}}$ on the coset space $G/L_{f}.$
 Thus there is one$-$to$-$one correspondence
 \be
(\mu_{G},\mu_{L_{f}}) \longleftrightarrow \mu_{G/L_{f}},
 \ee
between the pairs of measures $(\mu_{G},\mu_{L_{f}})$  and the measures $\mu_{G/L_{f}}$ on the coset spaces $G/L_{f}.$
Moreover, the Theorem gives explicitly $\mu_{G/L_{f}}$, and it describes
 how $\mu_{G/L_{f}}$ is constructed from the measures $\mu_{G}$ and $\mu_{L_{f}}.$






%This Theorem also
%gives explicitly the $G$$-$invariant  Haar measure on the orbit  $Gf$,
%%where $f$ is an arbitrary representative point of the orbit  $Gf$,
%and it describes in detail how
%they are
%it is constructed.
%This Theorem is a generalization of the corresponding statement which
%obviously holds when $G$ and $L_{f}$  are finite groups.



We note that the projection from the Haar measure on $G$ to the
$G$$-$invariant  Haar measure on the orbit $Gf \simeq G/L_{f}$ is
obvious when $G$ and $L_{f}$ are finite groups and the Theorem in question
generalizes beautifully this  obvious fact.
The sought
$G$$-$invariant measures on the orbits $Gf \simeq G/L_{f}$,
where $G=SL(2,R)$, and $L_{f}=SO(2)$, or $L_{f}=C_{n}$, where $n$ is even,
are obtained immediately by  applying this Theorem.
%is an immediate consequence of this Theorem.
%This Theorem
%describes explicitly the $G$$-$invariant  Haar measure on the orbits $Gf.$
Details are given in \cite{Melas}.







%(Existence and uniqueness of Haar measure) Let {G} be a %{\sigma}-compact locally compact Hausdorff group. Then there %exists a left-invariant %Haar measure {\mu} on {G}. %Furthermore, this measure is unique up to scalars: if {\mu, %\nu} are two left-invariant Haar measures on {G}, then %{\nu = %\lambda \mu} for some scalar {\lambda>0}.

%Similarly if “left-invariant” is replaced by “right-invariant” %throughout. (However, we do not claim that every %left-invariant Haar measure is automatically right-invariant, %or vice versa.)







%"A.Weil, L'integration dans les groupes topologiques et ses %applications, Paris, 1938"


%https://terrytao.wordpress.com/2011/09/27/254a-notes-3-haar-measure-and-the-peter-weyl-theorem/























































































































































































































































































































































































































































































































































































































































































































































%\noindent
%\bf (ii$_{\alpha}$) \normalfont In \cite{Melas} it is proved that the construction of a unique (up to a constant factor)
%$G$$-$invariant measure on the orbit $01 \equiv G/L_{\phi}$, $L_{\phi} = K = SO(2)$, necessitates the
%construction of a $G$$-$invariant measure on $G$, and the construction of a $K$$-$invariant measure on
%$K$. A $\;  G$$-$invariant measure on
%%We start by constructing an $\; \mathcal %G$$-$invariant measure on
%%$\;\mathcal G=G \times G, \;$ $\; G={\rm S}{\rm %L}(2,R). \;$
%% A $\;G$$-$invariant measure  $\; \mu_{G} \;$ on $\; %G={\rm S}{\rm
%%L}(2,R) \;$ is well known. Indeed, in \cite{gel}, %p.214$-$215 an
%%invariant measure on $\;{\rm S}{\rm L}(2,C)\;$ is %explicitly
%%constructed. With a similar construction one can %obtain an $\; G$$-$
%%invariant measure on $\;{\rm S}{\rm L}(2,R).\;$ So, if
%$$G={\rm S}{\rm L}(2,R)= \left \{ \left(
%\begin{array}{cc}
%a  & b \\
%c   &  d
%\end{array}
%\right), \quad a,b,c,d \in R, \quad ad-bc=1 \right \}, \;$$
%%invariant measure on $\;{\rm S}{\rm L}(2,R)\;$
%is given by \begin{equation}
%\label{lyhgbhgblyuitg7lyugk} {\rm d}g= \frac{{\rm d}a \wedge {\rm
%d}b \wedge {\rm d}c}{a}. \end{equation} $\; G$$-$invariant measure means
%$\; {\rm d}g={\rm d}(gg_{{\rm o}})={\rm d} (g_{{\rm o}}g), \
%g_{{\rm o}} \in {\rm S}{\rm L}(2,R). \;$ A $K$$-$invariant measure on $K$ is given by
%the 1$-$form ${\rm d} \theta$ where $\theta$ is the usual angular coordinates which covers $P_{1}(R) \simeq S^{1}$.
%\newline

%\noindent
%\bf (ii$_{\beta}$) \normalfont The orbits $02 \equiv G/L_{\phi}$,      $L_{\phi}=C_{n}$, where $n$ is even,  can be endowed with the %$G$$-$invariant measure on $G$ given in case $01$. Indeed, for a given ''little group'' $L_{\phi}$,  the orbit
%$02 \equiv G/L_{\phi}$ is the space of orbits of the right action
%$T : G \times L_{\phi} \rightarrow G$
% of the group $L_{\phi}$ on $G$
%given by
%\begin{equation}
%\label{action}
%g \star c := g \cdot c,
%\end{equation}
%where $g \in G$ and $c \in L_{\phi}$.
%Thus the action $T$ denoted by $\star$ is identical to the group multiplication in $G$. Since the group $ L_{\phi}$ is finite and since the %action (\ref{action}) is fixed point free the coset spaces $G/L_{\phi}$ inherit the measure on $G$.

%\emph{}

This completes the necessary information in order to construct the induced representations of $B(2,1)$.
As already stated, it is shown in \cite{Melas} that
when $B(2,1)$ is equipped with the Hilbert topology the inducing construction is exhaustive notwithstanding the fact that $B(2,1)$ is not locally compact.
 To conclude, in this paper \it all \normalfont IRS
%the continuous, unitary, irreducible %representations
of $B(2, 1)$ have been constructed in the Hilbert topology.







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%\noalign{\smallskip}\hline
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%\end{table}


%\begin{acknowledgements}
%If you'd like to thank anyone, place your comments here
%and remove the percent signs.
%\end{acknowledgements}

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%
%\bibitem{RefJ}
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%Author, Article title, Journal, Volume, page numbers (year)
%% Format for books
%\bibitem{RefB}
%Author, Book title, page numbers. Publisher, place (year)
%% etc
%Ας αδράξωμεν τα γεννητικά μας όργανα και ας αναφωνήσωμεν ομοθύμως και στεντορείως: `εις τους όρχεις ημών`

%\bibitem{Bondi} Bondi H,  Van Der Berg %M G J and  Metzner A W K 1962
%%\emph{Gravitational waves in general %relativity VII. Waves from
%%axi-symmetric isolated sources}.
%$Proc. \ R. \ Soc. \ Lond. \ A$ %\textbf{269}  21

\bibitem{Bondi}Bondi, H.,  Van Der Berg, M. G. J.,  Metzner A. W. K.:
%\emph{
Gravitational waves in general relativity VII. Waves from
axi$-$symmetric isolated sources.
%}.
Proc. R. Soc. Lond. A.
\textbf{269},  21$-$51 (1962)













%\bibitem{Sachs1} Sachs R K 1962
%%\emph{Gravitational waves in general %relativity VIII. Waves in
%%asymptotically flat space-time}.
% $Proc. \ R. \ Soc. \ Lond. \ A$ %\textbf{270} 103

\bibitem{Sachs1}Sachs, R. K.:
%\emph{
Gravitational waves in general relativity VIII. Waves in
asymptotically flat space$-$time.
%}.
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\bibitem{Pen1}Penrose, R:
%\emph{
Asymptotic Properties of Fields and Space$-$Times.
%}.
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\bibitem{Pen2}Penrose, R:
Relativistic symmetry groups, in: Group theory
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\bibitem{Pen3}
Penrose, R.,  Rindler, W.:   Spinors and spacetime, Vol. 2.
Cambridge Univ. Press, Cambridge (1986)







%\bibitem{Wigner} Wigner E 1939
%%\emph{On unitary represenations of %the Inhomogeneous Lorentz
%%Group}.
%$Annals \ of \ Mathematics $ %\textbf{40} 149

\bibitem{Wigner}Wigner, E.:
%\emph{
On unitary represenations of the Inhomogeneous Lorentz
Group.
%}.
Annals of Mathematics \textbf{40}, 149$-$204 (1939)










%\bibitem{mac3} McCarthy P J  1972
%%\emph{Representations of the %Bondi$-$Metzner$-$Sachs
%%group \quad   I. Determination of the %representations}.
%$Proc. \ R. \
%Soc. \ Lond. \ A.$ \textbf {330} 517

\bibitem{mac3}McCarthy, P. J.:
%\emph{
    Representations of the Bondi$-$Metzner$-$Sachs
group
%\quad
I. Determination of the representations.
%}
Proc. R. Soc. Lond. A.
\textbf {330}, 517$-$535 (1972)















%\bibitem{mac8} McCarthy P J 1972
%%\emph{Structure
%%of the  Bondi-Metzner-Sachs group .}
% $J. \ Math. \ Phys.$ \textbf {13} %1837

\bibitem{mac8}McCarthy, P. J.:          %    \emph{
    Structure
of the  Bondi$-$Metzner$-$Sachs group.
%}
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%\bibitem{mac4} McCarthy P J 1973
%%\emph{Representations of the %Bondi-Metzner-Sachs
%%group \quad   II. Properties and %classification of the  of the
%%representations}.
%$Proc. \ R. \ Soc. \ Lond. \ A $ %\textbf {333} 317









\bibitem{mac4}McCarthy, P. J.:    %    \emph{
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group
%\quad
II. Properties and classification of the  of the
representations.
%}
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%\bibitem{mac5}  McCarthy P J  and  %Crampin M  1973
%%\emph{Representations of the
%%Bondi$-$Metzner$-$Sachs group \quad   %III. Poincare spin
%%multiplicities and irreducibility}.
%$Proc. \ R. \ Soc. \ Lond. \ A.$ %\textbf
%{335} 301

\bibitem{mac5}McCarthy, P. J.,  Crampin, M.:
%    \emph{
    Representations of the
Bondi$-$Metzner$-$Sachs group
%$\quad
III. Poincare spin multiplicities and
irreducibility.
%}.
Proc. R. Soc. Lond. A. \textbf {335}, 301$-$311 (1973)












%\bibitem{mac6} McCarthy P J 1975
%%%\emph{The  Bondi-Metzner-Sachs group %in
%%%the nuclear topology}.
% $Proc. \ R. \ Soc. \ Lond. \ A$ %\textbf {343} 489

\bibitem{mac6}McCarthy, P. J.:      %    \emph{
    The  Bondi$-$Metzner$-$Sachs group in
the nuclear topology.
%}.
 Proc. R. Soc. Lond. A. \textbf {343}, 489$-$523 (1975)










%\bibitem{mac7} McCarthy P J 1978
%%%\emph{Lifting of projective %representations
%%%of the  Bondi-Metzner-Sachs group in %the nuclear topology.}
%$ Proc. \ R. \ Soc. \ Lond. \ A$ %\textbf {358} 141

\bibitem{mac7}McCarthy, P. J.:      %    \emph{
    Lifting of projective representations
of the  Bondi$-$Metzner$-$Sachs group in the nuclear topology.
%}
 Proc. R. Soc. Lond. A. \textbf {358}, 141$-$171 (1978)













%\bibitem{Crampin2}  McCarthy P J  and  %Crampin M 1974
%%\emph{Physical significance of the %topology of the
%%Bondi-Metzner-Sachs group }.
%$Phys. \ Rev. \ Lett.$ \textbf {33}
%547








\bibitem{Crampin2}McCarthy, P. J.,
%\&
Crampin, M.:
%\emph{
Physical significance of
the topology of the Bondi$-$Metzner$-$Sachs group.
%}.
 Phys. Rev. Letters. \textbf {33}, 547$-$550 (1974)






%\bibitem{mac9} McCarthy, P. J., Crampin, M.:
%%%%%\emph{
%Physical significance of the topology of the
%Bondi-Metzner-Sachs group.
%%}.
%%$
%Phys. Rev.  Lett.
%%$
%\textbf {29}, 547$-$550   (1974)




\bibitem{Mac}McCarthy, P. J.:
%\emph{
    Asymptotically flat Space$-$Times and Elementary Particles.
    %}.
Phys. Rev. Lett. \textbf {29}, 817$-$819
(1972)


\bibitem{Mackey0}Mackey, G. W.:
%\emph{
    Induced Representations of Locally Compact Groups I.
    %}.
Annals of Mathematics \textbf {55}, 101$-$139
(1952)

\bibitem{Mackey00}Mackey, G. W.:
%\emph{
    Induced Representations of Locally Compact Groups II. The
    Frobenius Reciprocity Theorem
    %}.
Annals of Mathematics \textbf {58}, 193$-$221
(1953)










\bibitem{Mackey} Mackey, G. W.:
%\emph{
Induced representations of groups and quantum mechanics.
%}
Benjamin, New York, Amsterdam and Torino (1968)
















\bibitem{Mackey1} Mackey, G. W.: 1955
%\emph{
The  theory of group representations.
%}
The University of Chicago Press, Chicago (1955)













\bibitem{Simms}Simms, D. J.:
%\emph{
Lie groups and quantum mechanics.
%}
%(Bonn notes
%)
Springer, Berlin, Heidelberg and New York   (1968)













\bibitem{Isham} Isham, C. J.:
%\emph{
Topological and quantum aspects of quantum theory
%}
in Relativity groups and topology. North-Holland, Amsterdam (1984)















%in
%Dewitt B.S. \& Stora R. (Editors), Relativity groups and topology
%II, p. 1059  Elsevier

\bibitem{Piard1}  Piard, A.:
%\emph{
UNITARY REPRESENTATIONS OF SEMIDIRECT PRODUCT GROUPS WITH INFINITE DIMENSIONAL ABELIAN NORMAL SUBGROUP.
%}
Rep.  Math.  Phys. \textbf{11}, 259$-$278 (1977)



\bibitem{Piard2}  Piard, A.:
%%\emph{
REPRESENTATIONS OF THE BONDI$-$METZNER$-$SACHS GROUP
WITH THE HILBERT TOPOLOGY.
%}
Rep.  Math.  Phys. \textbf{11}, 279$-$283 (1977)












\bibitem{mac1}McCarthy, P. J.:
Real and Complex Asymptotic Symmetries in Quantum Gravity, Irreducible Representations, Polygons, Polyhedra, and the A, D, E Series.
    %\emph{
Phil.Trans. R.  Soc. Lond. A.
%    }
    \textbf
{338}, 271$-$299 (1992)






















\bibitem{Mel1}  Melas, E.: The BMS group and generalized gravitational instantons.
 J. Math. Phys. \textbf{45} \normalfont 3,  996$-$1002 (2004)




\bibitem{macMel}McCarthy, P. J.,  Melas, E.:
%    \emph{
    On irreducible representations of the ultrahyperbolic
BMS group.
%}.
Nuc. Phys. B. \textbf {653}, 369$-$399 (2003)





\bibitem{Mel5} Melas, E.: Construction of the irreducibles of $B(2, 2)$. J.  Phys.  A:  Math.  Gen.  \textbf{39}, 3341$-$3366 (2006)


\bibitem{Melas1}  Melas, E.:
%\emph{
Representations of the Ultrahyperbolic BMS
HB. I. General Results
%}
(unpublished)



\bibitem{Mel6} Melas, E.: Representations of the ultrahyperbolic BMS group HB. II. Determination of the representations induced from infinite ``little groups''. (2013) arXiv:1312.0532v1 (2013)

\bibitem{Mel7} Melas, E.: Representations of the ultrahyperbolic BMS group HB. III. Determination of the representations induced from finite ``little groups''.  arXiv:1402.1428v1 (2014)


\bibitem{Mel8} Melas, E.: BMS groups in three space$-$time dimensions. $B(2,1)$ and its irreducible representations.
NEB$-$17 RECENT DEVELOPMENTS IN GRAVITY, SEPTEMBER 19$-$22, 2016, MYKONOS, GREECE (Accepted for publication in the Proceedings of the Conference).



\bibitem{ass}Ashtekhar, A.,  Hansen, R.O.:
%    \emph{
A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and
conserved quantities at spatial infinity
%}.
J. Math. Phys. \textbf{45} \normalfont 19,  1542$-$1566 (1978)


\bibitem{shaw}  Awada, M. A., Gibbons G. W.,   Shaw,  W. T.: Conformal supergravity, twistors, and
the super$-$BMS Group  Annals Phys. \textbf{171}, 52$-$107 (1986)




\bibitem{IshKak0} Isham, C. J.,   Kakas, A. C.: A group theoretical approach to the canonical quantisation of gravity. I. Construction of the canonical group.
    Class. Quantum Grav. \bf{1}, \normalfont 621$-$632 (1984)


\bibitem{IshKak} Isham, C. J.,   Kakas, A. C.: A group theoretical approach to the canonical quantisation of gravity. II. Unitary representations of the canonical group.
    Class. Quantum Grav. \bf{1}, \normalfont 633$-$650 (1984)


\bibitem{Bar3} Barnich, G., Oblak, B.:
Notes on the BMS group in three dimesnions: I. Induced representations
arXiv:1403.5803v3 [hep$-$th]


\bibitem{Bar4} Barnich, G., Oblak, B.:
Notes on the BMS group in three dimesnions: II. Coadjoint representation
arXiv:1502.00010v3 [hep$-$th]




























\bibitem{Hol} Hollands, S., Ishibashi, A.: Asymptotic flatness at null infinity in higher
dimensional gravity. arXiv:hep$-$th/0311178v2


\bibitem{Wald} Hollands, S., Wald, R. W.: Conformal Null Infinity Does Not Exist for Radiating Solutions in Odd Spacetime Dimensions. arXiv:gr$-$qc/0407014v1






\bibitem{Tan1} Tanabe, K., Tanahashi, N., Shiromizu, T.: On asymptotic structure at null infinity in five dimensions. arXiv:0909.0426v2 [gr$-$qc]



\bibitem{Tan2} Tanabe, K., Tanahashi, N., Shiromizu, T.: Angular momentum at null infinity in five dimensions. arXiv:1010.1664v2 [gr$-$qc]


\bibitem{Tan3} Tanabe, K., Kinoshita, S., Shiromizu, T.: Asymptotic flatness at null infinity in arbitrary dimensions. arXiv:1104.0303v2 [gr$-$qc]



\bibitem{Bram} Bramson, B. D.: Physics in Cone Space
in Asymptotic Structure of Space$-$Time, ed. F. P. Esposito and
L. Witten. Plenum Press, New York and London (1977)





\bibitem{Ger} Geroch, R.P.: Asymptotic Structure of Space$-$Time
in Asymptotic Structure of Space$-$Time, ed. F. P. Esposito and
L. Witten. Plenum Press, New York and London (1977)








\bibitem{Bij} Ashtekhar, A., B\.{i}\u{c}\'{a}k, J., Schm\.{i}dt, B. G.:   Asymptotic structure of symmetry reduced general relativity. arXiv:gr$-$qc/9608042v1


\bibitem{Bar1} Barnich, G., Troessaert, C.:
Symmetries of asymptotically flat 4 dimensional
spacetimes at null infinity revisited. arXiv:1104.0303v2 [gr$-$qc]


\bibitem{Bar2} Barnich, G., Troessaert, C.:
Aspects of the BMS/CFT correspondence. arXiv:1001.1541v2 [hep$-$th]







\bibitem{Str}   Strominger,  Α.:
Black hole entropy from near$-$horizon microstates.
JHEP {\bf 02}, 009 (1998) arXiv:hep$-$th/9712251



\bibitem{Str1} Guica,  M., Hartman, T., Song,  W., Strominger, A.:
The Kerr/CFT Correspondence.
Phys. Rev. D  {\bf 80}, 124008 (2009) arXiv:hep$-$th/0809.4266


\bibitem{Graham}  Fefferman, C., Graham  C. R.:
Conformal Invariants  in Élie Cartan et les Mathématiques d’Aujourd’hui Astérisque numéro hors série (Lyon 1984) 95$–$116 (1985)

\bibitem{Aha} Aharony, O., Gubser, S. S., Maldacena, J., H. Ooguri, H., Oz, Y.: Large
N Field Theories,
String Theory and Gravity.
Phys. Rept. {\bf 323}, 183$-$386 (2000) arXiv:hep$-$th/9905111 


\bibitem{Boer} J. de Boer, Solodukhin, S. N.:
A holographic reduction of Minkowski space$-$time
Nucl. Phys. B {\bf 665}, 545$-$593 (2003) arXiv:hep$-$th/0303006


\bibitem{Sol} Solodukhi, S. N.: Reconstructing Minkowski Space$-$Time. arXiv:hep$-$th/0405252v2



\bibitem{Melas}  Melas, E.:
%\emph{
On irreducible representations of the Bondi$-$Metzner$-$Sachs group in three space$-$time dimensions  in the Hilbert topology
%}
(unpublished)

\bibitem{Rudin}  Rudin, W.:  %    \emph{
    Functional Analysis.
%    }
    McGraw$-$Hill Inc., New York and London (1991)

\bibitem{Bour}  Bourbaki, N.: Measure de Haar. Hermann, Paris (1963)




%\bibitem{Mel}  Melas E
%%\emph{
%Representations of the
%Ultrahyperbolic BMS group $ \mathcal U \mathcal B \mathcal B(2,2)$. I. General Results
%%}
%(unpublished)


\bibitem{Frob} Frobenius, G. Sitz. d. K\"{o}n. Preuss. Akad. p.501 (1898)


%\bibitem{Cantoni} Cantoni, V.:
%%\emph{
%On the representations of the Bondi$-$Metzner$-$Sachs group.
%%}
%Ph.D. Thesis, University of London (1967)





\bibitem{Barut} Barut, A. O., Raczka, R.:
THEORY OF GROUP REPRESENTATIONS AND APPLICATIONS. Polish Scientific Publishers,
Warszawa (1980)


\bibitem{Cantoni} Cantoni, V.:
%\emph{
On the representations of the Bondi$-$Metzner$-$Sachs group.
%}
Ph.D. Thesis, University of London (1967)



\bibitem{Greiner}Greiner, W.:
%\emph{
CLASSICAL MECHANICS.
%}
%(Bonn notes
%)
Springer$-$Verlag  Berlin, Heidelberg and New York   (2004)

\bibitem{Bourbaki} Bourbaki, N.: Elements of Mathematics.
Integration II. Chapters 7$-$9. Springer$-$Verlag  Berlin, Heidelberg (2007)


\bibitem{Haar}
Haar, A.: Der Massbegriff in der Theorie der kontinuierlichen Gruppen. Ann. of Math. \bf{2}, \normalfont 147$-$169  (1933)


%p.214 215
\bibitem{Lang}
Lang, S.: $SL_{\rm{2}}\rm{(R)}.$ Springer$-$Verlag  Berlin, Heidelberg, New York, and Tokyo   (1985)



\bibitem{Gel}
Gel’fand, I. M.   et al.: Generalized functions, vol.5. Academic
Press  New York and London (1966)


































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to create references. Consult the Instructions % for authors for reference list style. % %
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{Bondi Bondi H, Van Der Berg %M G J and Metzner A W K 1962 %%Gravitational waves in general %relativity VII. Waves from %%axi-symmetric isolated sources. %$Proc. \ R. \ Soc. \ Lond. \ A$ %269 21
1703.05980.extracted_bib
{BondiBondi, H., Van Der Berg, M. G. J., Metzner A. W. K.: % Gravitational waves in general relativity VII. Waves from axi$-$symmetric isolated sources. %. Proc. R. Soc. Lond. A. 269, 21$-$51 (1962) %
1703.05980.extracted_bib
{Sachs1 Sachs R K 1962 %%Gravitational waves in general %relativity VIII. Waves in %%asymptotically flat space-time. % $Proc. \ R. \ Soc. \ Lond. \ A$ %270 103
1703.05980.extracted_bib
{Sachs1Sachs, R. K.: % Gravitational waves in general relativity VIII. Waves in asymptotically flat space$-$time. %. Proc. R. Soc. Lond. A.270, 103$-$126 (1962)
1703.05980.extracted_bib
{Pen1Penrose, R: % Asymptotic Properties of Fields and Space$-$Times. %. Phys. Rev. Lett. 10, 66 (1963)
1703.05980.extracted_bib
{Pen2Penrose, R: Relativistic symmetry groups, in: Group theory and nonlinear problems. Ed. A.O. Barut. Reidel, Dordrecht (1976)
1703.05980.extracted_bib
{Pen3 Penrose, R., Rindler, W.: Spinors and spacetime, Vol. 2. Cambridge Univ. Press, Cambridge (1986) %
1703.05980.extracted_bib
{Wigner Wigner E 1939 %%On unitary represenations of %the Inhomogeneous Lorentz %%Group. %$Annals \ of \ Mathematics $ %40 149
1703.05980.extracted_bib
{WignerWigner, E.: % On unitary represenations of the Inhomogeneous Lorentz Group. %. Annals of Mathematics 40, 149$-$204 (1939) %
1703.05980.extracted_bib
{mac3 McCarthy P J 1972 %%Representations of the %Bondi$-$Metzner$-$Sachs %%group \quad I. Determination of the %representations. %$Proc. \ R. \ %Soc. \ Lond. \ A.$ 330 517
1703.05980.extracted_bib
{mac3McCarthy, P. J.: % Representations of the Bondi$-$Metzner$-$Sachs group %\quad I. Determination of the representations. % Proc. R. Soc. Lond. A. 330, 517$-$535 (1972) %
1703.05980.extracted_bib
{mac8 McCarthy P J 1972 %%Structure %%of the Bondi-Metzner-Sachs group . % $J. \ Math. \ Phys.$ 13 %1837
1703.05980.extracted_bib
{mac8McCarthy, P. J.: % Structure of the Bondi$-$Metzner$-$Sachs group. % J. Math. Phys. 13, 1837$-$1842 (1972) %
1703.05980.extracted_bib
{mac4 McCarthy P J 1973 %%Representations of the %Bondi-Metzner-Sachs %%group \quad II. Properties and %classification of the of the %%representations. %$Proc. \ R. \ Soc. \ Lond. \ A $ %333 317
1703.05980.extracted_bib
{mac4McCarthy, P. J.: % Representations of the Bondi$-$Metzner$-$Sachs group %\quad II. Properties and classification of the of the representations. % Proc. R. Soc. Lond. A. 333, 317$-$336 (1973) %
1703.05980.extracted_bib
{mac5 McCarthy P J and %Crampin M 1973 %%Representations of the %%Bondi$-$Metzner$-$Sachs group \quad %III. Poincare spin %%multiplicities and irreducibility. %$Proc. \ R. \ Soc. \ Lond. \ A.$ %%{335 301
1703.05980.extracted_bib
{mac5McCarthy, P. J., Crampin, M.: % Representations of the Bondi$-$Metzner$-$Sachs group %$\quad III. Poincare spin multiplicities and irreducibility. %. Proc. R. Soc. Lond. A. 335, 301$-$311 (1973) %
1703.05980.extracted_bib
{mac6 McCarthy P J 1975 %%%The Bondi-Metzner-Sachs group %in %%%the nuclear topology. % $Proc. \ R. \ Soc. \ Lond. \ A$ %343 489
1703.05980.extracted_bib
{mac6McCarthy, P. J.: % The Bondi$-$Metzner$-$Sachs group in the nuclear topology. %. Proc. R. Soc. Lond. A. 343, 489$-$523 (1975) %
1703.05980.extracted_bib
{mac7 McCarthy P J 1978 %%%Lifting of projective %representations %%%of the Bondi-Metzner-Sachs group in %the nuclear topology. %$ Proc. \ R. \ Soc. \ Lond. \ A$ %358 141
1703.05980.extracted_bib
{mac7McCarthy, P. J.: % Lifting of projective representations of the Bondi$-$Metzner$-$Sachs group in the nuclear topology. % Proc. R. Soc. Lond. A. 358, 141$-$171 (1978) %
1703.05980.extracted_bib
{Crampin2 McCarthy P J and %Crampin M 1974 %%Physical significance of the %topology of the %%Bondi-Metzner-Sachs group . %$Phys. \ Rev. \ Lett.$ 33 %547
1703.05980.extracted_bib
{Crampin2McCarthy, P. J., %\& Crampin, M.: % Physical significance of the topology of the Bondi$-$Metzner$-$Sachs group. %. Phys. Rev. Letters. 33, 547$-$550 (1974) %
1703.05980.extracted_bib
{mac9 McCarthy, P. J., Crampin, M.: %%%%% %Physical significance of the topology of the %Bondi-Metzner-Sachs group. %%. %%$ %Phys. Rev. Lett. %%$ %29, 547$-$550 (1974)
1703.05980.extracted_bib
{MacMcCarthy, P. J.: % Asymptotically flat Space$-$Times and Elementary Particles. %. Phys. Rev. Lett. 29, 817$-$819 (1972)
1703.05980.extracted_bib
{Mackey0Mackey, G. W.: % Induced Representations of Locally Compact Groups I. %. Annals of Mathematics 55, 101$-$139 (1952)
1703.05980.extracted_bib
{Mackey00Mackey, G. W.: % Induced Representations of Locally Compact Groups II. The Frobenius Reciprocity Theorem %. Annals of Mathematics 58, 193$-$221 (1953)
1703.05980.extracted_bib
{Mackey Mackey, G. W.: % Induced representations of groups and quantum mechanics. % Benjamin, New York, Amsterdam and Torino (1968)
1703.05980.extracted_bib
{Mackey1 Mackey, G. W.: 1955 % The theory of group representations. % The University of Chicago Press, Chicago (1955)
1703.05980.extracted_bib
{SimmsSimms, D. J.: % Lie groups and quantum mechanics. % %(Bonn notes %) Springer, Berlin, Heidelberg and New York (1968)
1703.05980.extracted_bib
{Isham Isham, C. J.: % Topological and quantum aspects of quantum theory % in Relativity groups and topology. North-Holland, Amsterdam (1984) %in %Dewitt B.S. \& Stora R. (Editors), Relativity groups and topology %II, p. 1059 Elsevier
1703.05980.extracted_bib
{Piard1 Piard, A.: % UNITARY REPRESENTATIONS OF SEMIDIRECT PRODUCT GROUPS WITH INFINITE DIMENSIONAL ABELIAN NORMAL SUBGROUP. % Rep. Math. Phys. 11, 259$-$278 (1977)
1703.05980.extracted_bib
{Piard2 Piard, A.: %% REPRESENTATIONS OF THE BONDI$-$METZNER$-$SACHS GROUP WITH THE HILBERT TOPOLOGY. % Rep. Math. Phys. 11, 279$-$283 (1977)
1703.05980.extracted_bib
{mac1McCarthy, P. J.: Real and Complex Asymptotic Symmetries in Quantum Gravity, Irreducible Representations, Polygons, Polyhedra, and the A, D, E Series. % Phil.Trans. R. Soc. Lond. A. % 338, 271$-$299 (1992)
1703.05980.extracted_bib
{Mel1 Melas, E.: The BMS group and generalized gravitational instantons. J. Math. Phys. 45 \normalfont 3, 996$-$1002 (2004)
1703.05980.extracted_bib
{macMelMcCarthy, P. J., Melas, E.: % On irreducible representations of the ultrahyperbolic BMS group. %. Nuc. Phys. B. 653, 369$-$399 (2003)
1703.05980.extracted_bib
{Mel5 Melas, E.: Construction of the irreducibles of $B(2, 2)$. J. Phys. A: Math. Gen. 39, 3341$-$3366 (2006)
1703.05980.extracted_bib
{Melas1 Melas, E.: % Representations of the Ultrahyperbolic BMS HB. I. General Results % (unpublished)
1703.05980.extracted_bib
{Mel6 Melas, E.: Representations of the ultrahyperbolic BMS group HB. II. Determination of the representations induced from infinite ``little groups''. (2013) arXiv:1312.0532v1 (2013)
1703.05980.extracted_bib
{Mel7 Melas, E.: Representations of the ultrahyperbolic BMS group HB. III. Determination of the representations induced from finite ``little groups''. arXiv:1402.1428v1 (2014)
1703.05980.extracted_bib
{Mel8 Melas, E.: BMS groups in three space$-$time dimensions. $B(2,1)$ and its irreducible representations. NEB$-$17 RECENT DEVELOPMENTS IN GRAVITY, SEPTEMBER 19$-$22, 2016, MYKONOS, GREECE (Accepted for publication in the Proceedings of the Conference).
1703.05980.extracted_bib
{assAshtekhar, A., Hansen, R.O.: % A unified treatment of null and spatial infinity in general relativity. I. Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity %. J. Math. Phys. 45 \normalfont 19, 1542$-$1566 (1978)
1703.05980.extracted_bib
{shaw Awada, M. A., Gibbons G. W., Shaw, W. T.: Conformal supergravity, twistors, and the super$-$BMS Group Annals Phys. 171, 52$-$107 (1986)
1703.05980.extracted_bib
{IshKak0 Isham, C. J., Kakas, A. C.: A group theoretical approach to the canonical quantisation of gravity. I. Construction of the canonical group. Class. Quantum Grav. 1, \normalfont 621$-$632 (1984)
1703.05980.extracted_bib
{IshKak Isham, C. J., Kakas, A. C.: A group theoretical approach to the canonical quantisation of gravity. II. Unitary representations of the canonical group. Class. Quantum Grav. 1, \normalfont 633$-$650 (1984)
1703.05980.extracted_bib
{Bar3 Barnich, G., Oblak, B.: Notes on the BMS group in three dimesnions: I. Induced representations arXiv:1403.5803v3 [hep$-$th]
1703.05980.extracted_bib
{Bar4 Barnich, G., Oblak, B.: Notes on the BMS group in three dimesnions: II. Coadjoint representation arXiv:1502.00010v3 [hep$-$th]
1703.05980.extracted_bib
{Hol Hollands, S., Ishibashi, A.: Asymptotic flatness at null infinity in higher dimensional gravity. arXiv:hep$-$th/0311178v2
1703.05980.extracted_bib
{Wald Hollands, S., Wald, R. W.: Conformal Null Infinity Does Not Exist for Radiating Solutions in Odd Spacetime Dimensions. arXiv:gr$-$qc/0407014v1
1703.05980.extracted_bib
{Tan1 Tanabe, K., Tanahashi, N., Shiromizu, T.: On asymptotic structure at null infinity in five dimensions. arXiv:0909.0426v2 [gr$-$qc]
1703.05980.extracted_bib
{Tan2 Tanabe, K., Tanahashi, N., Shiromizu, T.: Angular momentum at null infinity in five dimensions. arXiv:1010.1664v2 [gr$-$qc]
1703.05980.extracted_bib
{Tan3 Tanabe, K., Kinoshita, S., Shiromizu, T.: Asymptotic flatness at null infinity in arbitrary dimensions. arXiv:1104.0303v2 [gr$-$qc]
1703.05980.extracted_bib
{Bram Bramson, B. D.: Physics in Cone Space in Asymptotic Structure of Space$-$Time, ed. F. P. Esposito and L. Witten. Plenum Press, New York and London (1977)
1703.05980.extracted_bib
{Ger Geroch, R.P.: Asymptotic Structure of Space$-$Time in Asymptotic Structure of Space$-$Time, ed. F. P. Esposito and L. Witten. Plenum Press, New York and London (1977)
1703.05980.extracted_bib
{Bij Ashtekhar, A., B\.{i\u{c\'{ak, J., Schm\.{idt, B. G.: Asymptotic structure of symmetry reduced general relativity. arXiv:gr$-$qc/9608042v1
1703.05980.extracted_bib
{Bar1 Barnich, G., Troessaert, C.: Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited. arXiv:1104.0303v2 [gr$-$qc]
1703.05980.extracted_bib
{Bar2 Barnich, G., Troessaert, C.: Aspects of the BMS/CFT correspondence. arXiv:1001.1541v2 [hep$-$th]
1703.05980.extracted_bib
{Str Strominger, Α.: Black hole entropy from near$-$horizon microstates. JHEP {02, 009 (1998) arXiv:hep$-$th/9712251
1703.05980.extracted_bib
{Str1 Guica, M., Hartman, T., Song, W., Strominger, A.: The Kerr/CFT Correspondence. Phys. Rev. D {80, 124008 (2009) arXiv:hep$-$th/0809.4266
1703.05980.extracted_bib
{Graham Fefferman, C., Graham C. R.: Conformal Invariants in Élie Cartan et les Mathématiques d’Aujourd’hui Astérisque numéro hors série (Lyon 1984) 95$–$116 (1985)
1703.05980.extracted_bib
{Aha Aharony, O., Gubser, S. S., Maldacena, J., H. Ooguri, H., Oz, Y.: Large N Field Theories, String Theory and Gravity. Phys. Rept. {323, 183$-$386 (2000) arXiv:hep$-$th/9905111
1703.05980.extracted_bib
{Boer J. de Boer, Solodukhin, S. N.: A holographic reduction of Minkowski space$-$time Nucl. Phys. B {665, 545$-$593 (2003) arXiv:hep$-$th/0303006
1703.05980.extracted_bib
{Sol Solodukhi, S. N.: Reconstructing Minkowski Space$-$Time. arXiv:hep$-$th/0405252v2
1703.05980.extracted_bib
{Melas Melas, E.: % On irreducible representations of the Bondi$-$Metzner$-$Sachs group in three space$-$time dimensions in the Hilbert topology % (unpublished)
1703.05980.extracted_bib
{Rudin Rudin, W.: % Functional Analysis. % McGraw$-$Hill Inc., New York and London (1991)
1703.05980.extracted_bib
{Bour Bourbaki, N.: Measure de Haar. Hermann, Paris (1963) %
1703.05980.extracted_bib
{Mel Melas E %% %Representations of the %Ultrahyperbolic BMS group $ \mathcal U \mathcal B \mathcal B(2,2)$. I. General Results %% %(unpublished)
1703.05980.extracted_bib
{Frob Frobenius, G. Sitz. d. K\"{on. Preuss. Akad. p.501 (1898) %
1703.05980.extracted_bib
{Cantoni Cantoni, V.: %% %On the representations of the Bondi$-$Metzner$-$Sachs group. %% %Ph.D. Thesis, University of London (1967)
1703.05980.extracted_bib
{Barut Barut, A. O., Raczka, R.: THEORY OF GROUP REPRESENTATIONS AND APPLICATIONS. Polish Scientific Publishers, Warszawa (1980)
1703.05980.extracted_bib
{Cantoni Cantoni, V.: % On the representations of the Bondi$-$Metzner$-$Sachs group. % Ph.D. Thesis, University of London (1967)
1703.05980.extracted_bib
{GreinerGreiner, W.: % CLASSICAL MECHANICS. % %(Bonn notes %) Springer$-$Verlag Berlin, Heidelberg and New York (2004)
1703.05980.extracted_bib
{Bourbaki Bourbaki, N.: Elements of Mathematics. Integration II. Chapters 7$-$9. Springer$-$Verlag Berlin, Heidelberg (2007)
1703.05980.extracted_bib
{Haar Haar, A.: Der Massbegriff in der Theorie der kontinuierlichen Gruppen. Ann. of Math. 2, \normalfont 147$-$169 (1933) %p.214 215
1703.05980.extracted_bib
{Lang Lang, S.: $SL_{\rm{2\rm{(R).$ Springer$-$Verlag Berlin, Heidelberg, New York, and Tokyo (1985)
1703.05980.extracted_bib
{Gel Gel’fand, I. M. et al.: Generalized functions, vol.5. Academic Press New York and London (1966)