Equivariant indices of vector fields and 1-forms
W.~Ebeling and S.~M.~Gusein-Zade \thanks{Partially supported by the DFG (Mercator fellowship, Eb 102/8--1), the Russian government grant 11.G34.31.0005, RFBR--13-01-00755, and NSh--4850.2012.1. %% and Simons--IUM fellowship. Keywords: singularities, vector fields, 1-forms, finite group action, index, Burnside ring. AMS 2010 Math. Subject Classification: 14B05, 58K45, 58A10, 58E40, 19A22. }
Equivariant versions of the radial index and of the GSV-index of a vector field or a 1-form on a singular variety with an action of a finite group are defined. They have values in the Burnside ring of the group. Poincar\'e--Hopf type theorems for them are proven and some of their properties are described.
1307.2054.tex
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\title{Equivariant indices of vector fields and 1-forms}
\author{W.~Ebeling and S.~M.~Gusein-Zade
\thanks{Partially supported by the DFG (Mercator fellowship, Eb 102/8--1),
the Russian government grant 11.G34.31.0005, RFBR--13-01-00755,
and NSh--4850.2012.1. %% and Simons--IUM fellowship.
Keywords: singularities, vector fields, 1-forms, finite group action, index, Burnside ring.
AMS 2010 Math. Subject Classification: 14B05, 58K45, 58A10, 58E40, 19A22.
}
}
%% \date{\today}
\date{}

\begin{document}
\selectlanguage{english}

\maketitle

\begin{abstract}
Equivariant versions of the radial index
and of the GSV-index of a vector field or a 1-form on a singular variety with an action of a finite group are defined. They have values in the Burnside ring of the group. Poincar\'e--Hopf type theorems for them are proven and some of their properties are described.
\end{abstract}

%%%%%%%%%%%%%%%%%%%%%%%
\section*{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%
The classical notion of the index of an isolated singular point (zero) of a vector field
has several generalizations to vector fields and 1-forms on singular varieties.
In particular, there were defined the GSV-index of a vector field on an isolated hypersurface
or complete intersection singularity \cite{GSV, SS}, the Euler obstruction of a 1-form \cite{MacPh},
the homological index of a vector field \cite{G-MM} or of a 1-form \cite{EGS}, the radial index of a vector
field or of a 1-form, \dots

The radial index of an isolated singular point of a (stratified) vector field or of a 1-form on a singular variety
was defined in \cite{GD, Sur}, following earlier ideas of \cite{MHSch} and \cite{MMJ}.
Initially it was defined for  vector fields and 1-forms on analytic varieties. However without any difference
one can consider the radial index on closed (real) subanalytic varieties (as noticed in \cite{D}). The sum of the
radial indices of a vector field or of  a 1-form with isolated singular points on a compact variety is equal
to the Euler characteristic of the variety. Therefore the (radial) index can be considered as a localization
of the Euler characteristic (on a singular variety).

There are several equivariant versions of the Euler characteristic for spaces with actions of
a finite group $G$. First, one can consider the Euler characteristic of the quotient space.
One can define an equivariant Euler characteristic as an element of the representation ring
of the group $G$: see, e.g., \cite{TtD, Wall1}. There exists the so-called orbifold Euler
characteristic (with values in the ring $\ZZ$ of integers) offered by physicists \cite{Vafa, HH}
and its higher order generalizations \cite{AS, BF}. However a more general concept is the
equivariant Euler characteristic defined as an element of the Burnside ring $B(G)$ of the group $G$
\cite{Verdier, TtD}. The other versions of the Euler characteristic mentioned above are
specializations of this one.

This makes reasonable to try to define indices of singular points of vector fields and of 1-forms
on $G$-varieties as elements of the Burnside ring $B(G)$
%%%%%%%%%%%%%%%%%
so that they will be localizations of the equivariant Euler characteristic.
%%%%%%%%%%%%%%%%%
Their specializations will give other
equivariant versions of the indices, say, as an element of the ring of representations or an
orbifold version (with values in $\ZZ$). For vector fields on (smooth) $G$-manifolds this was
done in \cite{Luck}.

In this paper we define equivariant versions with values in the Burnside ring of the radial index
and of the GSV-index, prove Poincar\'e--Hopf type theorems for them and describe some of their
properties.

In Section~\ref{sect1} we recall the concepts of the radial index and of the GSV-index in the usual
(non-equivariant) setting. In Section~\ref{sect2} we give definitions of the equivariant Euler
characteristic, of the orbifold Euler characteristic and of its higher order generalizations.
In Section~\ref{sect3} we discuss an equivariant version of the radial index. Section~\ref{sect4}
is devoted to an equivariant version of the GSV-index.

%%%%%%%%%%%%%%%%%%%%%%%
\section{Indices of vector fields and 1 forms}\label{sect1}
%%%%%%%%%%%%%%%%%%%%%%%
Let $(V,0)$ be a germ of a closed (real) subanalytic variety. We assume $(V,0)$ to be embedded into $(\RR^N, 0)$.
Let $V=\bigcup_{i=1}^q V_i$ be a subanalytic Whitney stratification of $V$.
A (continuous) {\em stratified vector field} on $V=\bigcup_{i=1}^q V_i$
is a vector field such that, at each point $p$ of $V$, it is tangent
to the stratum containing $p$ \cite{BSS}.

Let $B_\eps$ be a closed ball (in the ambient space $\RR^N$) around the origin
(small enough so that the sphere $S_\eps=\partial B_\eps$ intersects all
the strata $V_i$ transversally). Let $L=V\cap S_\eps$ be the link of $V$.
There exists a diffeomorphism $h$ of $V\cap B_\eps$ with the cone
$CL=L\times[0,1]/L\times\{0\}$ over $L$ being the identity on $L=L\times\{1\}$.
(A diffeomorphism of stratified varieties is a homeomorphism which respects
the stratification and is a diffeomorphism on each stratum.) A vector field $Y$
on $(V,0)$ is called {\em radial} if, for a certain diffeomorphism $h$,
it is identified with the vector field $\frac{\partial{\ }}{\partial t}$ on $CL$.

Let $p\in V$, let $V_{(p)}=V_i$ be the stratum containing $p$, $\dim V_i=k$,
and let $Y$ be a stratified vector field on $V$ in a neighbourhood
of the point $p$. Let $N_i$ be a normal slice of $V$ to the
stratum $V_i$ at the point $p$. There exists a diffeomorphism $h$
from a neighbourhood of $p$ in $V$ to the product $U_i(p)\times N_i$, where
$U_i(p)$ is a neighbourhood of $p$ in $V_i$, which is the identity on $U_i(p)$.
The vector field $Y$ is called a {\em radial extension} of the vector field
$Y_{\vert V_i}$ if there exists a diffeomorphism $h$ which identifies $Y$
with the vector field on $U_i(p)\times N_i$ of the form
$(Y_{\vert V_i},0)+(0, Y^{\rm rad}_{N_i})$, where $Y^{\rm rad}_{N_i}$
is a radial vector field on $N_i$ \cite{BSS}.

Let $X$ be a stratified vector field on $(V,0)$ with an isolated singular point
(zero) at the origin. One can show that there exists a (continuous) $G$-invariant
stratified vector field $\widetilde{X}$ on $V$ such that:
\begin{itemize}
\item[(1)] the vector field $\widetilde{X}$ coincides with $X$ on a neighbourhood
of the intersection of $V$ with the sphere $S_\eps=\partial B_\eps$
of a small radius $\eps$ around the origin;
\item[(2)] the vector field has a finite number of singular points (zeros);
\item[(3)] in a neighbourhood of each singular point $p\in V\cap B_\eps$, $p\in V_i$,
the vector field $\widetilde{X}$ is a radial extension of
its restriction to the stratum $V_i$.
\end{itemize}

The {\em radial index} $\indrad(X;V,0)$ of the vector field $X$
on $V$ at the origin is 
$$
\sum_{{p}\in {\rm Sing}\widetilde{X}} 
{\rm ind\,}(\widetilde{X}_{\vert V_{(p)}};V_{(p)},p)\,,
$$
where ${\rm ind\,}(\widetilde{X}_{\vert V_{(p)}},V_{(p)},p)$ is the usual index of the 
restriction of the vector field $\widetilde{X}$ to the smooth manifold $V_{(p)}$.

Now let $\omega$ be a (continuous) 1-form on $(V,0)$ (i.e.\ the restriction to $V$ of a 1-form
defined in a neighbourhood of $0$ in $\RR^N$). Let $V=\bigcup_{i=1}^q V_i$ be a subanalytic Whitney stratification of $V$. A point $p\in V$ is a {\em singular point} of $\omega$
if the restriction of $\omega$ to the stratum $V_{(p)}$ containing $p$ vanishes at the point $p$.
A 1-form $\eta$ on $(V,0)$ is called {\em radial} if its restriction to any (parametrized) analytic
curve $\varphi:(\RR_{\ge0},0)\to (V,0)$ is positive on the vector $\varphi_*\frac{\partial\ }{\partial t}$ 
in a punctured neighbourhood of zero. As an example, one can take $d \Vert \cdot \Vert^2$, where 
$\Vert \cdot \Vert$ is the Euclidean norm in $\RR^N$.

For a point $p$, $p\in V_i=V_{(p)}$, $\dim V_{(p)}=k$, there exists a (local) analytic
diffeomorphism $h:(\RR^N,\RR^k,0)\to (\RR^N, V_{(p)},p)$. A 1-form $\eta$ is called a
{\em radial extension} of the 1-form $\eta_{\vert V_{(p)}}$ if there exists a diffeomorphism
$h$ which identifies $\eta$ with the restriction to $V$ of the 1-form 
$\pi_1^* \eta_{\vert V_{(p)}}+\pi_2^* \eta^{rad}_{\RR^{N-k},0}$, where $\pi_1$ and $\pi_2$ 
are the projections of $\RR^N$ to $\RR^k$ and $\RR^{N-k}$ respectively,
$\eta^{rad}_{\RR^{N-k},0}$ is a radial 1-form on $(\RR^{N-k},0)$.

%\begin{remark}
% Note that the definition of a radial extension of a vector field cannot be formulated in similar terms
% since, in contrast to 1-forms, one cannot restrict a vector field defined on the 
% ambient space $\RR^N$  to $V$. Therefore one uses a product/cone-like structure of $V$ in a %neighbourhood of $p$ which is  guaranteed by the Whitney  conditions.
%\end{remark}

For a 1-form $\omega$ on $(V,0)$ with an isolated singular point at the origin there exists
a 1-form $\widetilde{\omega}$ on $V$ which possesses the obvious analogues of the properties (1)--(3)
of the vector field $\widetilde{X}$ above. The {\em radial index} of $\omega$ is 
$$
\indrad(\omega;V,0)=
\sum_{{p}\in {\Sing}\widetilde{\omega}} 
\ind(\widetilde{\omega}_{\vert V_{(p)}};V_{(p)},p)\,.
$$

There is a one-to-one correspondence between complex valued and real valued 1-forms on a complex analytic variety.
The real 1-form $\alpha$ corresponding to a complex 1-form $\omega$ is just its real part: $\alpha={\rm Re}\omega$.
The 1-form $\omega$ can be restored from $\alpha$ by the equation $\omega(u)=\alpha(u)-\alpha(iu)i$
($u$ is a tangent vector to the ambient space). Therefore, 
for a complex 1-form on a complex analytic variety, its radial index is defined as the radial index of its real part. For a complex 1-form on a $d$-dimensional complex analytic manifold, this index differs from the usual one by the sign $(-1)^d$.

The radial index of a vector field or of a 1-form is defined as the sum of the indices
of an appropriate deformation of it on the strata (smooth manifolds). The sum is the same for
a stratification and for a partition of it. This implies that the radial index (and also
its equivariant version below) does not depend on the stratification and thus is well-defined.
(For a vector field one can consider the intersection of two stratifications.
For a 1-form one can consider the minimal Whitney stratification of the variety.)

Let $(V,0)$ be a germ of a subanalytic variety, let $V=\bigcup_{i=1}^q V_i$ be a (subanalytic) Whitney stratification
of $V$, and let $X$ or $\omega$ be a stratified vector field or a 1-form on $(V,0)$ respectively with an isolated
singular point at the origin. One can show that the radial index $\indrad(X; V,0)$ or $\indrad(\omega,V,0)$ can be distributed between the strata $V_i$ in the following sense. The index is the number of singular points of an
appropriate deformation $\widetilde{X}$ or $\widetilde{\omega}$ of $X$ or $\omega$ respectively counted with multiplicities. 

\begin{proposition}\label{prop-distr}
The number of singular points of the vector field $\widetilde{X}$ or of a 1-form $\widetilde{\omega}$
on a fixed stratum $V_i$ does not depend on the choice of the vector field $\widetilde{X}$ or of the 1-form $\widetilde{\omega}$ respectively (and therefore only depends on $X$ or $\omega$ respectively).
\end{proposition}

\begin{proof}
We shall write the proof in terms of vector fields. The proof for 1-forms is the same. We shall follow this scheme in the sequel as well.

Let $\widetilde{X}_1$ and $\widetilde{X}_2$ be two deformations of $X$ with the necessary properties.
Let $W_i$ be the complement in $V_i$ of a small tubular neighbourhood of its boundary $\partial V_i$.
The intersection $W_i\cap B_\eps$ (after an appropriate smoothing at $\partial W_i \cap S_\eps$) is
a manifold with boundary. The vector fields $\widetilde{X}_1$ and $\widetilde{X}_2$ coincide on the part
$W_i\cap S_\eps$ of the boundary of $W_i$. On the remaining part these vector fields are homotopic,
in the class of non-vanishing vector fields, to vector fields pointing inside $W_i$ and therefore are homotopic
to each other (in the same class). This implies that the numbers of the singular points of the vector fields
$\widetilde{X}_1$ and $\widetilde{X}_2$ on $W_i$ counted with multiplicities coincide.

Another argument for the statement can be the following one.
The closure $\overline{V}_i$ of the stratum $V_i$ and its boundary $\overline{V}_i\setminus V_i$
are closed subanalytic varieties. The numbers of the singular points of the vector field $\widetilde{X}$
on $\overline{V}_i$ and on $\overline{V}_i\setminus V_i$ (counted with multiplicities) are equal
to the indices $\indrad(X;\overline{V}_i,0)$ and $\indrad(X;{\overline{V}_i\setminus V_i,0})$ respectively
(and therefore are well-defined). The number of the singular points of $\widetilde{X}$
on ${V}_i$ is the difference 
$$
\indrad(X;\overline{V}_i,0)-\indrad(X;{\overline{V}_i\setminus V_i,0}).
$$
\end{proof}

This means that the radial index can be defined for each stratum $V_i$ so that
$$
\indrad(X;V,0)=\sum_{i=1}^q \indrad(X; V_i,0)\,.
$$
Moreover the radial index can be defined as a linear map from the set of constructible,
integer valued functions on $V$ constant on each stratum $V_i$ to $\ZZ$. (The radial index
$\indrad(X;V,0)$ is the image of the constant function $1$.)

There is also the notion of the GSV-index of a vector field or a 1-form on a complex analytic complete intersection singularity. 
We shall discuss this notion (and therefore its $G$-equivariant version below)  in the generality considered in \cite{BSS}.
Let $f_1, \ldots, f_k$ be holomorphic
function germs on $(\CC^n,0)$ defining a complete intersection singularity 
$(V,0)=\{f_1=\ldots=f_k=0\}$. We assume that a neighbourhood of the origin in  $\CC^n$ permits a Whitney stratification adapted to V and satisfying the Thom $a_f$ condition. This holds, in particular, if $(V,0)$ is an
isolated complete intersection singularity (ICIS) or if it is a hypersurface, i.e.\ $k=1$. Let $F$ be the map
$F=(f_1, \ldots, f_k): (\CC^n,0)\to(\CC^k,0)$ and let $\Delta\subset (\CC^k,0)$ be the discriminant of $F$.
For $0<\delta\ll\eps$ small enough the restriction of the map $F$ to $B_{\eps}^{2n}$ is a locally trivial fibration
over $B_{\delta}^{2k}\setminus\Delta$ ($B_{\eps}^{2n}$ and $B_{\delta}^{2k}$ are the closed balls
of radii $\eps$ and $\delta$ around the origin in $\CC^n$ and in $\CC^k$ respectively.).
For $\underline{\delta}=(\delta_1,\ldots, \delta_k)\in B_{\delta}^{2k}\setminus\Delta$, the fibre
$M_{\underline{\delta}}={F}^{-1}(\underline{\delta})\cap B_{\eps}^{2n}$ of this fibration is a non-singular complex manifold with the boundary $\partial M_{\underline{\delta}}=F^{-1}(\underline{\delta})\cap S_{\eps}^{2n-1}$~---
the {\em Milnor fibre} of $F$ or of $(V,0)$ (the set $S_{\eps}^{2n-1}=\partial B_{\eps}^{2n}$
is the sphere of radius $\eps$ around the origin in $\CC^n$).

Let $X$ be a stratified vector field on $(V,0)$ with an isolated singular point at the origin. In particular,
the restriction of $X$ to a neighbourhood of $V\cap S_{\eps}^{2n-1}$ has no zeros.
In \cite{BSS} it is explained how one can construct a vector field $\widehat{X}$ on a neighbourhood of the boundary
$\partial M_{\underline{\delta}}$ of the Milnor fibre approximating $X$ on a neighbourhood of
$V\cap S_{\eps}^{2n-1}$. Two approximations of this sort
are close to each other and therefore homotopic to each other in the class of non-vanishing vector
fields. Let $\widetilde{X}$ be an extension of  $\widehat{X}$ to the Milnor fibre $M_{\underline{\delta}}$
with isolated singular points.

The GSV-{\em index} of the vector field $X$ is
\begin{equation*}\label{GSV}
\indGSV(X;V,0)=
\sum_{p \in\Sing \widetilde{X}}
\ind(\widetilde{X}; M_{\underline{\delta}},p)\,.
\end{equation*}

%%%%%%%%%%%%%%%%%%%%%%%
\section{Burnside rings and Euler characteristics}\label{sect2}
%%%%%%%%%%%%%%%%%%%%%%%

Let $G$ be a finite group. Let $\Conjsub G$ be the set of conjugacy classes of subgroups of $G$.
A $G$-set is a set with an action of the group $G$.
A $G$-set is {\em irreducible} if the action of $G$ on it is transitive.
Isomorphism classes of irreducible $G$-sets are in one-to-one correspondence with
the elements of $\Conjsub G$: to the conjugacy class $[H]$ containing
a subgroup $H\subset G$ one associates the isomorphism class $[G/H]$
of the $G$-set $G/H$. The {\em Burnside ring} $B(G)$ of $G$ is the
Grothendieck ring of finite $G$-sets, i.e.\
the abelian group generated by the isomorphism classes of finite $G$-sets
modulo the relation $[A\amalg B]=[A]+[B]$ for finite $G$-sets $A$ and $B$. The multiplication
in $B(G)$ is defined by the 
%% Cartesian
cartesian product. As an abelian group, $B(G)$
is freely generated by the isomorphism classes of irreducible $G$-sets,
i.e.\ each element of $B(G)$ can be written in a unique way as 
$\sum\limits_{[H]\in \Conjsub G}a_{[H]}[G/H]$ with $a_{[H]}\in\ZZ$.
The element $1$ in
the ring $B(G)$ is represented by the $G$-set $[G/G]$ consisting of one point (with the trivial $G$-action).

There is a natural homomorphism from the Burnside ring $B(G)$ to the ring $R(G)$
of representations of the group $G$ which sends a $G$-set $X$ to the (vector)
space of functions on $X$. If $G$ is cyclic, then this homomorphism is injective.
In general, it is neither injective nor surjective.

For a subgroup $H\subset G$ there are natural maps
$\mbox{R}_{H}^{G}: B(G)\to B(H)$ and $\mbox{I}_{H}^{G}: B(H)\to B(G)$.
The {\em restriction map} $\mbox{R}_{H}^{G}$ sends a $G$-set X to the same set considered with the $H$-action.
The {\em induction map} $\mbox{I}_{H}^{G}$ sends an $H$-set $X$ to the product $G\times X$ factorized
by the natural equivalence: $(g_1,
 x_1)\equiv (g_2, x_2)$ if there exists $g\in H$ such that
$g_2=g_1g$, $x_2=g^{-1}x_1$ with the natural (left) $G$-action. Both maps are group homomorphisms,
however the induction map $\mbox{I}_{H}^{G}$ is not a ring homomorphism. 

For an action of a group $G$ on a set $X$ and for a point $x\in X$, let $G_x=\{g\in G: gx=x\}$
be the isotropy group of the point $x$. For a subgroup $H\subset G$ let
$X^H=\{x\in X: Hx=x\}$ be the fixed point set of the subgroup $H$
and let $X^{(H)}=\{x\in X: G_x=H\}$ be the set of points with the isotropy group $H$.
For a conjugacy class $[H]\in \Conjsub G$, let
$X^{[H]}=\bigcup\limits_{K\in [H]}X^{K}$,
$X^{([H])}=\bigcup\limits_{K\in [H]}X^{(K)}$.

For a ``sufficiently good'' topological space $V$, say, a subanalytic variety,
its Euler characteristic is
$$
\chi(V)=\sum_i (-1)^i \dim H^i_c(V;\RR),
$$
where $H^i_c(V;\RR)$ are the cohomology groups with compact support. This Euler characteristic
is additive (in contrast to the one defined via the usual cohomology groups).
For quasi-projective complex analytic varieties both these Euler characteristics coincide. 

Let $G$ be a finite group acting on $V$.

\begin{definition}
 The {\em equivariant Euler characteristic} of $(V,G)$ is the element of the Burnside ring of $G$
 defined by
 $$
 \chi^G(V)=\sum_{{[H]}\in\Conjsub G}\chi(V^{([H])}/G)[G/H]
 $$
 (see, e.g., \cite{Verdier, TtD}). The {\em reduced equivariant Euler characteristic} of $(V,G)$
 is
 $$
 {\overline{\chi}}^G(V)=\chi^G(V)-[G/G]\,.
 $$
\end{definition}

\begin{remark} For a subgroup $H$ from a conjugacy class $[H]$, the $G$-action
 on $V$ induces an $N_G(H)$-action on $V^H$ ($N_G(H)$ is the normalizer $\{g\in G: g^{-1}Hg=H\}$ of $H$)
 and $V^{([H])}/G=V^{({H})}/N_G(H)$. Therefore
$$
\chi^G(V)=\sum\limits_{{[H]}\in\Conjsub G}\chi(V^{(H)}/N_G(H))[G/H] .
$$
\end{remark} 

\begin{remark}
 One can see that the natural homomorphism from $B(G)$ to the ring $R(G)$ of representations of $G$
described above sends the equivariant Euler characteristic $\chi^G(V)$ to the equivariant Euler characteristic $\chi_G(V)\in R(G)$ considered, e.g., in \cite{Wall1}.
\end{remark}

There is a natural homomorphism $\vert\cdot\vert$ from the Burnside ring $B(G)$ to $\ZZ$ sending a (virtual) $G$-set
$A$ to the number of elements $\vert A\vert$ of $A$. This homomorphism sends the equivariant Euler characteristic
$\chi^G(V)$ to the usual Euler characteristic $\chi(V)$.

The {\em orbifold Euler characteristic} of the pair $(V,G)$ is defined by
$$ 
\chi^{orb}(V,G)=\frac{1}{\vert G\vert}\sum_{(g,h):gh=hg} \chi(X^{\langle g,h\rangle}) 
$$ 
where $\langle g,h\rangle$ is the subgroup of $G$ generated by $g$ and $h$ (see \cite{Vafa, HH, AS}). 

There are {\em higher order Euler characteristics} of the pair $(V,G)$ defined by
$$ 
\chi^{(k)}(V,G)=\frac{1}{\vert G\vert}
\sum_{{{\bf g}\in G^{k+1}:}\atop{g_ig_j=g_jg_i}} \chi(X^{\langle g_0, g_1, \ldots, g_k\rangle}), \quad k\ge 0
$$ 
where ${\bf g}=(g_0,g_1, \ldots, g_k)$, $\langle g_0, g_1, \ldots, g_k\rangle$ is the subgroup 
generated by $g_0,g_1, \ldots, g_k$ (see \cite{AS, BF}). The zeroth order Euler characteristic $\chi^{(0)}(V,G)$ of the pair $(V,G)$ is nothing else but the Euler characteristic
of the quotient $V/G$, the first order Euler characteristic $\chi^{(1)}(V,G)$ of the pair $(V,G)$ is just the orbifold one.
All these Euler characteristics are additive functions on the Grothendieck ring of subanalytic $G$-varieties.

There are homomorphisms $r_G^{(k)}$ from $B(G)$ as an abelian group to $\ZZ$ which send the equivariant Euler
characteristic $\chi^G(V)$ to the higher order Euler characteristics $\chi^{(k)}(V,G)$, $k\ge 0$. The homomorphism
$r_G^{(k)}$ is defined by $r_G^{(k)}([G/H])=\chi^{(k)}([G/H],G)$. The homomorphism $r_G^{(0)}$ sends the class $[A]$
of a finite $G$-set to the number of $G$-orbits in $A$. For an abelian group $G$ one has
$\chi^{(k)}([G/H],G)=\vert H\vert^k$. Therefore one can say that the higher order Euler characteristics are reductions of the equivariant Euler characteristic which is a more general one. The homomorphism $r_G^{(k)}$ is not, in general, a ring homomorphism. It is a ring homomorphism for abelian $G$ and $k=1$ (i.e.\ for
the orbifold Euler characteristic).

Below we construct $G$-equivariant versions of some integer valued invariants as elements of the
Burnside ring $B(G)$. They possess the property that the ``number of elements'' homomorphism $\vert\cdot\vert$
sends them to the original invariants (i.e.\ for trivial $G$). Applying the homomorphism $r_G^{(k)}$ one gets an orbifold version of each invariant (for $k=1$)
and also higher order integral invariants. (This was used in \cite{GLM} to define orbifold versions of the Lefschetz number.)


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Equivariant radial index}\label{sect3}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
We introduce the notion of an equivariant radial index for vector fields. The necessary changes for 1-forms are clear.

Let $(V,0)$ be a germ of a closed real subanalytic variety with an action
of a finite group $G$. (We assume $(V,0)$ to be embedded into $(\RR^N, 0)$
and the $G$-action to be induced by an analytic action on a neighbourhood of
$(V,0)$ in $(\RR^N, 0)$. This essentially means that the action on $(\RR^N, 0)$
can be assumed to be linear.) 
Let $V=\bigcup_{i=1}^q V_i$ be a subanalytic Whitney
$G$-stratification of $V$. This means that each stratum $V_i$ is $G$-invariant,
the isotropy subgroups $G_p=\{g\in G: gp=p\}$ of all points $p$ of $V_i$ are conjugate to each other,
and the quotient of the stratum by the group $G$
is connected. (Such a stratification can be obtained from an arbitrary subanalytic Whitney
stratification of $(V,0)$ by taking intersections/unions of the strata and
of their transforms by the elements of $G$.)

Let $X$ be a $G$-invariant stratified vector field on $(V,0)$ with an isolated singular point
(zero) at the origin. One can show that there exists a (continuous) $G$-invariant
stratified vector field $\widetilde{X}$ on $V$ satisfying conditions (1)--(3) from Section~\ref{sect1}.

Let $A$ be the set (a $G$-set) of the singular points of the vector field
$\widetilde{X}$ on $V\cap B_\eps$ considered with the multiplicities equal to
the usual indices $\ind(\widetilde{X}_{\vert V_{(p)}}; V_{(p)},p)$ of the restrictions
of the vector field $\widetilde{X}$ to the corresponding strata (smooth manifolds).

\begin{definition}
 The {\em equivariant radial index} $\indrad^G(X; V,0)$ of the vector field $X$
 on $V$ at the origin is the class $[A]\in B(G)$ of the set $A$ of singular
 points of $\widetilde{X}$ with multiplicities.
\end{definition}

\begin{remark}
 One can rewrite the definition as
\begin{equation*}\label{Defrad}
\indrad^G(X; V,0)=\sum_{\overline{p}\in ({\Sing}\widetilde{X})/G} 
\ind(\widetilde{X}_{\vert V_{(p)}}; V_{(p)},p) [Gp]\,, 
\end{equation*}
 where $p$ is a point of the preimage of $\overline{p}$ under the canonical projection.
\end{remark}

%%%%%%%%%%%%%%%%%%%%%%
For a subgroup $H\subset G$, the vector field $X$ is $H$-invariant and one has
$\indrad^H(X; V,0)=R^G_H(\indrad^G(X; V,0))$.
%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%
The homomorphisms $r_G^{(k)}$ give "higher order Euler characteristic versions" of the equivariant index:
$\indrad^{G,(k)}(X; V,0)=r_G^{(k)}(\indrad^G(X; V,0))$. For $k=1$ one gets an orbifold index of $X$.
%%%%%%%%%%%%%%%%%%%%%%

For a stratum $V_i$ of the stratification,
let $[G_i]\in \Conjsub G$ be the conjugacy class of the isotropy subgroups of points
of $V_i$, and let $G_i$ be a representative of it. Singular points of the vector field $\widetilde{X}$
on the stratum $V_i$ are distributed between the orbits of the $G$-action, all of which (as $G$-sets)
are isomorphic to $[G/G_i]$. Therefore
\begin{equation}\label{equi1}
\indrad^G(X; V,0)= \sum_{i=1}^q \frac{\vert G_i\vert}{\vert G\vert}\ind(X;V_i,0) [G/G_i]
\end{equation}
(see Proposition~\ref{prop-distr}).
 
Let us write the equivariant index $\indrad^G(X; V,0)$ in terms of the usual (non-equivariant)
indices of the vector field $X$ on the fixed point sets $V^H$ of subgroups of $G$ or
on the ``fixed point sets'' $V^{[H]}=\bigcup\limits_{K\in [H]}V^K$
of conjugacy classes of subgroups of $G$.

Assume that
\begin{equation}\label{equi2}
 \indrad^G(X; V,0) = \sum_{[H]\in\Conjsub G} a_{[H]} [G/H]\,.
\end{equation}
One has
\begin{equation*}
 \indrad(X;V,0) = \sum_{[H]\in\Conjsub G} a_{[H]} \frac{\vert G\vert}{\vert H\vert} \,.
\end{equation*}
Equation~\ref{equi2} implies that
\begin{equation*}
 \indrad^G(X;V^{[H]},0) = \sum_{[K]\ge [H]} a_{[K]} [G/K]
\end{equation*}
and therefore
\begin{equation}\label{equi3}
\indrad(X;V^{[H]},0)  =
 \sum_{[K]\ge[H]} a_{[K]} \frac{\vert G\vert}{\vert K\vert}\,.
\end{equation}
Equation~\ref{equi3} can be rewritten as 
\begin{equation}\label{equi4}
\indrad(X;V^{[H]},0) =
 \sum_{[K]\in\Conjsub G} \zeta([H],[K])a_{[K]}
 \frac{\vert G\vert}{\vert K\vert}\,,
\end{equation}
 where
\begin{equation*}
 \zeta([H],[K])=\left\{ \begin{array}{cl} 0 & \mbox{for\ }[K]\not\ge[H],\\
 1 & \mbox{for\ }[K]\ge [H] \end{array} \right.
\end{equation*}
is the zeta function of the partially ordered set $\Conjsub G$. (One has $[K]\ge [H]$
if there exist representatives $K$ and $H$ of $[K]$ and $[H]$ respectively such that
$K\supset H$.)
The system~(\ref{equi4}) is a triangular system of linear equations with respect to $a_{[H]}$.
Its solution gives the following statement.

\begin{proposition}
One has 
\begin{equation*}
 \indrad^G(X; V,0) = \sum_{[H]\in\Conjsub G}\frac{\vert H\vert}{\vert G\vert}
\left( \sum_{[K]\in\Conjsub G}\mu([H], [K]) \indrad(X;V^{[K]},0) \right)[G/H]\,,
\end{equation*}
where $\mu([H],[K])$ is the Moebius function of the poset $\Conjsub G$ (see, e.g, \cite{Hall}).
\end{proposition}

In a similar way one has
\begin{equation}\label{equi5}
 \indrad(X;V^{H},0) =
 \sum_{K\ge H} a_{[K]} \frac{\vert N_G(K)\vert}{\vert K\vert}\,,
\end{equation}
where $N_G(K)$ is the normalizer $\{g\in G:g^{-1}Kg=K\}$ of the subgroup $K$.
This can be rewritten as 
\begin{equation}\label{equi6}
 \indrad(X;V^{H},0) =
 \sum_{K\in\Sub G} \zeta'(H,K)a_{[K]}
 \frac{\vert N_G(K)\vert}{\vert K\vert}\,,
\end{equation}
 where
%% \begin{equation*}
$\zeta'(H,K)$
%% =\left\{ \begin{array}{cl} 0\,, & \mbox{for\ }[K]\not\ge[H]\\
%% 1 & \mbox{for\ }[K]\ge[H] \end{array} \right.
%% \end{equation*}
is the zeta function of the partially ordered set $\Sub G$ of subgroups of $G$.
Thus one gets the following statement.

\begin{proposition} \label{propsubgroups}
One has 
\begin{equation*}
 \indrad^G(X;V,0) = \sum_{[H]\in\Conjsub G}\frac{\vert H\vert}{\vert N_G(H)\vert}
 \left(\sum_{K\in\Sub G}\mu'(H,K) \indrad(X;V^{K},0)\right)[G/H]\,,
\end{equation*}
where $\mu'(H,K)$ is the Moebius function of the poset $\Sub G$.
\end{proposition}

Let $V$ be a locally compact subanalytic $G$-variety %% with a subanalytic Witney stratification
and let $X$ be a $G$-invariant (stratified with respect to a stratification) vector field on $V$ with isolated
singular points. Let $p$ be a singular point of $X$. The germ $(V,p)$ caries an action of the isotropy subgroup $G_p$
of $p$. Therefore the index $\indrad^{G_p}(X;V,p)\in B(G_p)$ is defined.

\begin{definition}
The {\em $G$-equivariant index} $\indrad^G(X;V,Gp)$ of the vector field $X$ at the orbit $Gp$ is
 $I_{G_p}^G (\indrad^{G_p}(X;V,p)) \in B(G)$.
\end{definition}

One can see that the $G$-equivariant index of a vector field (and of a 1-form as well)
satisfies the law of conservation of number:
for a ($G$-invariant) deformation $\widetilde{X}$ of a vector field $X$ the sum of the $G$-equivariant
indices of $\widetilde{X}$ at the orbits in a neighbourhood of an orbit $Gp$ is equal to
$\indrad^G(X; V,Gp)$.

One has the following equivariant version of the Poincar\'e-Hopf Theorem.

\begin{theorem}
Let $V=\bigcup_{i=1}^q V_i$ be a compact subanalytic variety and let $X$ be a $G$-invariant stratified vector
field on $V$ with isolated singular points. Then the sum of the indices of the vector field $X$ over all singular orbits
is equal to the $G$-equivariant Euler characteristic of $V$:
$$
\sum_{\overline{p}\in (\Sing X)/G} \indrad^G(X;V,\pi^{-1}(\overline{p})) =\chi^G(V)\in B(G)
$$
($\pi:V\to V/G$ is the canonical projection).
\end{theorem}

An analogous statement holds for 1-forms.

\begin{proof}
Let $\widetilde{X}$ be a $G$-invariant deformation of $X$ which differs from $X$ only in neighbourhoods of
singular points and which satisfies
 the properties of the definition of the equivariant index. One has
\begin{eqnarray*}
\sum_{\overline{p}\in (\Sing X)/G} \indrad^G(X,V,\pi^{-1}(\overline{p})) &=&
\sum_{\overline{p}\in \Sing X/G} \indrad^G(\widetilde{X};V,\pi^{-1}(\overline{p})) \\
& = & 
\sum_{i=1}^q \sum_{p\in V_i}\frac{\vert G_i\vert}{\vert G\vert} \ind(\widetilde{X};V_i,p) [G/G_i]\,, \\
\chi^G(V) &= & \sum_{i=1}^q \frac{\vert G_i\vert}{\vert G\vert} \chi(V_i)[G/G_i]\,.
\end{eqnarray*}

As in the proof of Proposition~\ref{prop-distr}, let $W_i$ be the complement in $V_i$ of a small tubular neighbourhood
of its boundary $\partial V_i$. It can be regarded as a manifold with boundary. On the boundary 
$\partial W_i$ the restriction
$\widetilde{X}_{\vert W_i}$ of the vector field $\widetilde{X}$ is homotopic, in the
class of non-vanishing vector fields, to a vector field pointing inwards. Therefore 
$$
\sum_{p\in V_i}\ind(\widetilde{X};V_i,p)= \chi(W_i\setminus \partial W_i)=\chi(V_i)\,.
$$
(We use the fact that, on a compact manifold with boundary, the sum of the indices of the
singular points of a vector field
pointing inwards on the boundary is equal to the Euler characteristic of the interior of the manifold.)
\end{proof}

Let $(V,0)$ be a germ of a complex analytic variety of pure dimension $n$ and let $f:(V,0)\to(\CC,0)$ be a germ of a complex analytic function. The germ $f$ is the lifting $\overline{f}\circ \pi$ of an analytic function  germ
$\overline{f}:(V/G,0)\to(\CC,0)$ ($\pi:V\to V/G$ is the canonical projection).
The Milnor fibres $M_{{\rm Re} f}^+$ and $M_{{\rm Re} f}^-$ of ${\rm Re} f$ are $G$-homotopy equivalent to the Milnor fibre
$f^{-1}(\delta)\cap B_\eps$ of $f$ and one has $\indrad^G(df;V,0)=-\overline{\chi}^G(M_f)$.
The element $\mu^G_f=(-1)^{n-1}\overline{\chi}^G(M_f)\in B(G)$
can be regarded as an equivariant version of the Milnor number of $f$. The homomorphism $r_G^{(0)}$ from $B(G)$ to $\ZZ$ which sends $[G/H]$ to 1 maps the element $\mu^G_f$ to the Milnor number of the germ $\overline{f}$ on
$(V/G, 0)$ defined (for smooth $V/G$) in \cite{Wall1}.

Let $(V,0)$ be a germ of a closed subanalytic $G$-variety, let $(V/G,0)$ be the quotient by the $G$-action,
let $\pi:V\to V/G$ be the canonical projection and let $V=\bigcup\limits_{i=1}^q V_i$ and 
$V/G=\bigcup\limits_{i=1}^q V_i/G$ be Whitney stratifications of $V$ and $V/G$ respectively compatible with the
partitions corresponding to different orbit types. 
For a stratum $V_i$, $i=1,\ldots, q$, let $[G_i]$ be the conjugacy class of the isotropy subgroups of points of $V_i$
and let $G_i$ be a representative of it. 
Let $X$ be a $G$-invariant stratified vector field on $V$
with an isolated singular point at $0$. The image of $X$ under the map $\pi_*$ defines a stratified vector field
$\overline{X}$ on $V/G$ with an isolated singular point. 
One can easily see that a product/cone-like structure in a neighbourhood of a point $\overline{p}\in V_i/G$
defines a $G_p$-invariant product/cone-like structure in a neighbourhood of a point $p\in \pi^{-1}(\overline{p})$.
This implies that, if $\widetilde{X}$ is an appropriate (in the sense of the definition of the radial index) deformation of the vector
field $X$, then $\pi_*\widetilde{X}$ is an appropriate deformation of the vector
field $\overline{X}$. Therefore one has the following statement.

\begin{proposition} \label{prop-vect}
$$ 
\indrad^G(X;v,0)=\sum_{[H]\in\Conjsub G} \indrad(\overline{X};V^{([H])}/G, 0)[G/H]\,,
$$
where $\indrad(\overline{X};V^{([H])}/G, 0)$ is the (radial) index of the vector field $\overline{X}$ on the
non-closed variety $V^{([H])}/G$ defined in Section~\ref{sect1}.
\end{proposition}

Let $\omega$ be a 1-form on $V/G$ with an isolated singular point at $0$. The 1-form $\pi^*\omega$ on $V$
is $G$-invariant and has an isolated singular point at $0$. One can easily see that the following analogue
of Proposition~\ref{prop-vect} holds.

\begin{proposition}
$$
\indrad^G(\pi^*\omega; V,0)=\sum_{[H]\in\Conjsub G} \indrad(\omega;V^{([H])}/G, 0)[G/H]\,.
$$
\end{proposition}

\begin{example} Assume that $f:(V,0)\to(\RR,0)$ is a $G$-invariant germ of an analytic function on $(V,0)$
with an isolated critical point at $0$. (This means that the differential $df$ of $f$ has an isolated singular point at $0$.)
For a small enough positive $\eps$ the restriction of $f$ to $V\cap B_{\eps}$ is a locally trivial fibration over a punctured neighbourhood of zero in $\RR$, in general with different fibres over the positive and the negative
part. The fibres ($G$-varieties) $M_f^+$ and $M_f^-$ of this fibration can be called positive and negative Milnor fibres of $f$ respectively. One has the following generalization of \cite[Theorem 2]{EGS}.
\end{example}

\begin{proposition}\label{ind-mu-form}
$$
\indrad^G(df;V,0)=-\overline{\chi}^G(M_f^-)\,.
$$
\end{proposition}

For a ($G$-invariant) complex analytic function $f:(V,0)\to(\CC,0)$ on a germ of a complex analytic variety,
the Milnor fibres $M_f^{\pm}$ of its real part are homeomorphic to the product $M_f\times I$, where
$M_f=f^{-1}(\delta)\cap B_{\eps}$ is the Milnor fibre of $f$ ($0<\vert\delta\vert\ll\eps$ small enough),
$I$ is the segment $[0,1]$: see \cite[Proposition 2.A.3]{GM}. Therefore in this case 
$$
\indrad^G(df;V,0)=-\overline{\chi}^G(M_f)\,.
$$

\begin{example}\label{ex-dim}
Let the complex vector space $\CC^n$ be equipped with a (linear) $G$-action and let
$\omega=\sum\limits_{i=1}^n A_i(z)dz_i$ be a complex analytic $G$-invariant 1-form on $(\CC^n, 0)$
with an isolated singular point at the origin. For a subgroup $K\subset G$, let $n_K$ be the dimension of the
fixed point set $(\CC^n)^K$ of $K$ (a vector subspace). The (radial) index $\indrad(\omega; (\CC^n)^K,0)$
is $(-1)^{n_K}$ times the dimension of the space
$\Omega_{(\CC^n)^K,\, \omega}:=\Omega_{(\CC^n)^K,0}^{n_K}/(\Omega_{(\CC^n)^K,0}^{n_K-1}\wedge \omega_{\vert(\CC^n)^K})$, where
$\Omega_{(\CC^n)^K,0}^{m}$ is the vector space of germs of analytic differential $m$-forms on $((\CC^n)^K,0)$.
Together with Proposition~\ref{propsubgroups} this gives
$$
\indrad^G(\omega;\CC^n,0)=
$$
$$
\sum_{[H]\in\Conjsub G}\frac{\vert H\vert}{\vert N_G(H)\vert}
\left(\sum_{K\in {\Sub G}}(-1)^{n_K}\mu'(H,K)
\dim\Omega_{(\CC^n)^K,\,\omega}\right)
[G/H]\,.
$$
\end{example}

%%%%%%%%%%%%%%%%
\begin{example}
Let $f(z_1, \ldots, z_n)=\sum_{i=1}^n \prod_{j=1}^n z_i^{E_{ij}}$ be an invertible polynomial (i.e.\ $E_{ij} \in \ZZ$, $E_{ij} \geq 0$, $\det(E_{ij})\ne 0$) with an isolated critical point at the origin. Let 
$$
G_f=\{(\lambda_1, \ldots, \lambda_n)\in\CC^n:
f(\lambda_1 z_1, \ldots, \lambda_n z_n)=f(z_1, \ldots, z_n)\}
$$ 
be the group of diagonal symmetries of $f$.
The Berglund-H\"ubsch dual polynomial to $f$ is
$\widetilde{f}(z_1, \ldots, z_n)=\sum_{i=1}^n \prod_{j=1}^n z_i^{E_{ji}}$.
The group $G_{\widetilde{f}}$ of diagonal symmetries of $\widetilde{f}$ is (canonically) isomorphic
to $G_f^*={\rm Hom}(G,\CC^*)$ \cite{BLMS}. The result of \cite{BLMS} together with Proposition~\ref{ind-mu-form} says
that the indices $\indrad^{G_f}(df;\CC^n,0)$ and $\indrad^{G_{\widetilde{f}}}(d\widetilde{f};\CC^n,0)$
are Saito dual to each other in the sense of \cite{BLMS}.
Let $\overline{f}$ and $\overline{\widetilde{f}}$ be the functions on $\CC^n/G_f$ and
$\CC^n/G_{\widetilde{f}}$ corresponding to $f$ and $\widetilde{f}$ respectively. The Saito dual
elements of the Burnside rings $B(G_f)$ and $B(G_{\widetilde{f}})$ have the same images
under $r^{(0)}_{G_f}$ and $r^{(0)}_{G_{\widetilde{f}}}$ respectively. Therefore the indices
$\indrad^{G_f,(0)}(df;\CC^n,0)=\indrad(d\overline{f}; \CC^n/G_f,0)$ and
$\indrad^{G_{\widetilde{f}},(0)}(d\widetilde{f};\CC^n,0)=
\indrad(d\overline{\widetilde{f}}; \CC^n/G_{\widetilde{f}},0)$ (equal to the minus reduced Euler
characteristics of $M_f/G_f$ and $M_{\widetilde{f}}/G_{\widetilde{f}}$ respectively) coincide.
For a subgroup $H\subset G_f$ its dual  $H^T\subset G_{\widetilde{f}}$ is the kernel of the natural map $i^*: G_f^* \to H^*$ induced by the inclusion
$i : H \to G_f$. The result of \cite{MMJ2} says that the orbifold indices $\indrad^{H,(1)}(df;\CC^n,0)$ and $\indrad^{H^T,(1)}(d\widetilde{f};\CC^n,0)$ coincide.
\end{example}
%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Equivariant GSV-index}\label{sect4}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $(\CC^n,0)$ be endowed by a $G$-action. (Without loss of generality one may assume that the action is linear,
i.e.\ it is defined by a representation $G\to {\rm GL}(n)$.) Let $f_1, \ldots, f_k$ be $G$-invariant holomorphic
function germs on $(\CC^n,0)$ defining a complete intersection singularity 
$(V,0)=\{f_1=\ldots=f_k=0\}$. We assume that a neighbourhood of the origin in  $\CC^n$ permits a Whitney stratification adapted to V and satisfying the Thom $a_f$ condition.  

Let $X$ be a $G$-invariant stratified vector field on $(V,0)$ with an isolated singular point at the origin. In particular,
the restriction of $X$ to a neighbourhood of $V\cap S_{\eps}^{2n-1}$ has no zeros.
The vector fields $\widehat{X}$ and $\widetilde{X}$ described in Section~\ref{sect1} can be made $G$-invariant. (To get a $G$-invariant vector field, one can take an arbitrary one and take the mean over the group.)

\begin{definition}
The {\em $G$-equivariant GSV-index} of the vector field $X$ is
\begin{equation}\label{G-GSV}
\indGSV^G(X;V,0)=
\sum_{\overline{p}\in (\Sing \widetilde{X})/G}
\indrad^G (\widetilde{X}; M_{\underline{\delta}},\pi^{-1}(\overline{p}))\in B(G),
\end{equation}
where $\pi:M_{\underline{\delta}}\to M_{\underline{\delta}}/G$ is the canonical projection.
\end{definition}

It is easy to show that the right hand side of (\ref{G-GSV}) does not depend on the extension $\widetilde{X}$
of $\widehat{X}$ and therefore the equivariant GSV-index is well-defined.

A compact complex analytic $G$-variety $V$, all singular points of which are ICIS, has a natural smoothing
$\widetilde{V}$. (The manifold $V$ is in general $C^{\infty}$, not necessarily complex analytic.
It can be chosen to be complex analytic if $V$ is a projective complete intersection with
isolated singularities.) Let $X$ be a $G$-invariant vector field on $V$ with isolated singular points.
For $p\in \Sing X$, the $G$-equivariant GSV-index of $X$ at the orbit $Gp$ is defined as
$$
\indGSV^G(X; V,Gp):=I_{G_p}^G (\indGSV^{G_p}(X;V,p))\,.
$$
Since the GSV-index counts singular points of the vector field on the 
smoothing of the ICIS, one has the following version of the Poincar\'e--Hopf Theorem.

\begin{proposition}
The sum of the equivariant GSV-indices of the singular orbits of $X$ on $V$ is equal to the 
equivariant Euler characteristic of $\widetilde{V}$:
$$
\sum_{Gp\in(\Sing X)/G} \indGSV^G(X;V,Gp) = \chi^G(\widetilde{V})\in B(G)\,.
$$
\end{proposition}

A relation between the equivariant GSV-index and the radial one can be described in the following way.
The Milnor fibre $M_{\underline{\delta}}$ is a manifold with a $G$-action and therefore its equivariant
Euler characteristic $\chi^G(M_{\underline{\delta}})\in B(G)$ is defined. Let 
$\overline{\chi}^G(M_{\underline{\delta}})=\chi^G(M_{\underline{\delta}})-1$ be the reduced
equivariant Euler caracteristic of $M_{\underline{\delta}}$. (The element
$(-1)^{n-k} \overline{\chi}^G(M_{\underline{\delta}})\in B(G)$ can be regarded as an equivariant version
of the Milnor number of the ICIS $(V,0)$.)

\begin{proposition} \label{propGSV-rad}
One has
$$
\indGSV^G(X;V,0)=\indrad^G(X;V,0)+\overline{\chi}^G(M_{\underline{\delta}})\,.
$$
\end{proposition}

\begin{proof}
 This follows from the following facts: 1) both the GSV-index and the radial one 
 satisfy the law of conservation of number, and 2) for the radial vector field
 $X_{rad}$ on $(V,0)$ one has $\ind^G(X_{rad};V,0)=1$,
 $\indGSV^G(X_{rad};V,0)=\overline{\chi}^G(M_{\underline{\delta}})$.
\end{proof}

Analogous notions and statements exist for $G$-invariant 1-forms on  complete intersection singularities.

\begin{example}
 Let $(V,0)=\{f_1=\ldots=f_k=0\}$ be an ICIS. The equivariant Euler characteristic
 $\chi^G(M_{\underline{\delta}})$ of the Milnor fibre of $(V,0)$ is equal to the sum
 of the equivariant indices of the $G$-orbits of singular points on $M_{\underline{\delta}}$
 of a radial 1-form on $V$. Applying Proposition~\ref{propsubgroups} to each of the indices one gets:
 \begin{equation*}
 \chi^G(M_{\underline{\delta}}) = \sum_{[H]\in\Conjsub G}\frac{\vert H\vert}{\vert N_G(H)\vert}
 \left(\sum_{K\in\Sub G}\mu'(H,K) \chi(M_{\underline{\delta}})\right)[G/H]\,.
\end{equation*}
Since $\sum\limits_{K}\mu'(H,K)=\sum\limits_{K}\zeta(K,G)\mu'(H,K)=\delta(H,G)$, where $\delta(\cdot , \cdot)$ is the Kronecker delta function,  one has
 \begin{equation}\label{chi}
 \overline{\chi}^G(M_{\underline{\delta}}) = \sum_{[H]\in\Conjsub G}\frac{\vert H\vert}{\vert N_G(H)\vert}
 \left(\sum_{K\in\Sub G}\mu'(H,K) \overline{\chi}(M_{\underline{\delta}})\right)[G/H]\,.
\end{equation}
Let $\omega$ be a holomorphic $G$-invariant 1-form with an isolated singular point at the origin.
If the dimension of the fixed point set $(\CC^n)^K$ of a subgroup $K\subset G$ is greater than $k$, then
the GSV-index $\indGSV(\omega;(\CC^n)^K,0)$ is $(-1)^{n_K-k}$ times the dimension of the space
$\Omega_{V^K,\,\omega}=\Omega^{n_K-k}_{V^K,0}/(\omega\wedge \Omega^{n_K-k-1}_{V^K,0})$,
where $\Omega^{s}_{V^K,0}:=
\Omega^{s}_{(\CC^n)^K,0}/
(f_i\Omega^{s}_{(\CC^n)^K,0}, df_i\wedge \Omega^{s-1}_{(\CC^n)^K,0})$ \cite{MMJ, MZ}.

Propositions~\ref{propsubgroups}, \ref{propGSV-rad} and Equation~\ref{chi} imply that
%% \begin{equation*}
$$
\indGSV^G(\omega; V,0) = \sum_{[H]\in\Conjsub G}\frac{\vert H\vert}{\vert N_G(H)\vert}
 \left(\sum_{{K\in\Sub G}\atop{\dim{(\CC^K)^K} > k}}
 \mu(H,K) \dim \Omega_{V^K, \,\omega}\right)[G/H]\,.
$$
%% \end{equation*}
\end{example}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{10}
\bibitem{AS} M.~Atiyah, G.~Segal:
On equivariant Euler characteristics. 
J. Geom. Phys. {\bf 6}, no.4, 671--677 (1989). 

\bibitem{BF} J.~Bryan, J.~Fulman:
Orbifold Euler characteristics and the number of commuting $m$-tuples in the symmetric groups. 
Ann. Comb. {\bf 2}, no.1, 1--6 (1998). 

\bibitem{Vafa} L.~Dixon, J.~Harvey, C.~Vafa, E.~Witten:
Strings on orbifolds. Nucl. Phys. {\bf B 261}, 678--686 (1985), II. Nucl. Phys. {\bf B 274}, 285--314 (1986).

\bibitem{BSS} J.-P.~Brasselet, J.~Seade, T.~Suwa: Vector fields on singular varieties.
Lecture Notes in Mathematics, 1987. Springer-Verlag, Berlin, 2009.

\bibitem{D} N.~Dutertre: Radial index and Poincar\'e-Hopf index of 1-forms on semi-analytic sets.
Math. Proc. Cambridge Philos. Soc. {\bf 148}, no.2, 297--330 (2010).

\bibitem{MMJ} W.~Ebeling, S.M.~Gusein-Zade: Indices of 1-forms on an isolated complete
intersection singularity. Mosc. Math. J. {\bf 3}, no.2, 439--455 (2003).

\bibitem{GD} W.~Ebeling, S.M.~Gusein-Zade: Radial index and Euler obstruction of a 1-form
on a singular variety. Geom. Dedicata {\bf 113}, 231--241 (2005).

\bibitem{MZ} W.~Ebeling, S.~M.~Gusein-Zade: Quadratic forms for a 1-form on an isolated complete intersection singularity. Math. Z. {\bf 252}, 755--766 (2006).

\bibitem{Sur} W.~Ebeling, S.M.~Gusein-Zade: Indices of vector fields and 1-forms on singular varieties. In: Global aspects of complex geometry, 129--169,
Springer, Berlin, 2006.

\bibitem{BLMS} W.~Ebeling, S.M.~Gusein-Zade:
Saito duality between Burnside rings for invertible polynomials.
Bull. Lond. Math. Soc. {\bf 44}, no.4, 814--822 (2012).

\bibitem{MMJ2} W.~Ebeling, S.M.~Gusein-Zade:
Orbifold Euler characteristics for dual invertible polynomials.
Mosc. Math. J. {\bf 12}, no.1, 49--54, 214 (2012). 

\bibitem{EGS} W.~Ebeling, S.M.~Gusein-Zade, J.~Seade:
Homological index for 1-forms and a Milnor number for isolated
singularities. Int. J. of Math. \textbf{15}, 895--905 (2004).

\bibitem{GLM} S.M.~Gusein-Zade, I.~Luengo, A.~Melle-Hern\'andez:
On an equivariant version of the zeta function of a transformation.
arXiv: 1203.3344.

\bibitem{G-MM} X.~G\'omez-Mont: An algebraic formula
for the index of a vector field on a hypersurface with an isolated
singularity. J. Alg. Geom. \textbf{7}, 731--752 (1998).

\bibitem{GSV} X.~G\'omez-Mont, J.~Seade, A.~Verjovsky:
The index of a holomorphic flow with an isolated singularity. Math. Ann.
\textbf{291}, 737--751 (1991).

\bibitem{GM} M.~Goresky, R.~MacPherson:
Stratified Morse theory. Springer-Verlag, Berlin, 1988.

\bibitem{Hall} M.~Hall Jr.: Combinatorial theory. Second edition.
Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience Publication.
John Wiley \& Sons, Inc., New York, 1986.

\bibitem{HH} F.~Hirzebruch, Th.~H\"ofer:
On the Euler number of an orbifold.
Math. Ann. {\bf 286}, no.1-3, 255--260 (1990).

\bibitem{Luck} W.~L\"uck, J.~Rosenberg: The equivariant Lefschetz fixed point theorem
for proper cocompact $G$-manifolds. In: High-dimensional manifold topology, 322--361,
World Sci. Publ., River Edge, NJ, 2003.

\bibitem{MacPh} R.~MacPherson: Chern classes for
singular varieties. Annals of Math. \textbf{100}, 423--432 (1974).

\bibitem{MHSch} M.-H.~Schwartz: Classes caract\'eristiques
d\'efinies par une stratification d'une vari\'et\'e analytique complexe.
C. R. Acad. Sci. Paris S\'er. I Math. \textbf{260}, 3262--3264,
3535--3537 (1965)

\bibitem{SS} J.~A.~Seade, T.~Suwa: A residue formula for
the index of a holomorphic flow. Math. Ann. \textbf{304}, 621--634 (1996).

\bibitem{TtD} T.~tom Dieck: Transformation groups and representation theory.
Lecture Notes in Mathematics, 766, Springer, Berlin, 1979.

\bibitem{Verdier} J.-L.~Verdier: Caract\'eristique d'Euler-Poincar\'e. Bull. Soc. Math. France {\bf 101},
441--445 (1973).

\bibitem{Wall1} C.T.C.~Wall: A note on symmetry of singularities. Bull. London Math. Soc. {\bf 12},
no.3, 169--175 (1980).

\end{thebibliography}

\bigskip
\noindent Leibniz Universit\"{a}t Hannover, Institut f\"{u}r Algebraische Geometrie,\\
Postfach 6009, D-30060 Hannover, Germany \\
E-mail: ebeling@math.uni-hannover.de\\

\medskip
\noindent Moscow State University, Faculty of Mechanics and Mathematics,\\
Moscow, GSP-1, 119991, Russia\\
E-mail: sabir@mccme.ru

\end{document}







1307.2054.extracted_bib
\begin{thebibliography}{10}
\bibitem{AS} M.~Atiyah, G.~Segal:
On equivariant Euler characteristics. 
J. Geom. Phys. {\bf 6}, no.4, 671--677 (1989). 

\bibitem{BF} J.~Bryan, J.~Fulman:
Orbifold Euler characteristics and the number of commuting $m$-tuples in the symmetric groups. 
Ann. Comb. {\bf 2}, no.1, 1--6 (1998). 

\bibitem{Vafa} L.~Dixon, J.~Harvey, C.~Vafa, E.~Witten:
Strings on orbifolds. Nucl. Phys. {\bf B 261}, 678--686 (1985), II. Nucl. Phys. {\bf B 274}, 285--314 (1986).

\bibitem{BSS} J.-P.~Brasselet, J.~Seade, T.~Suwa: Vector fields on singular varieties.
Lecture Notes in Mathematics, 1987. Springer-Verlag, Berlin, 2009.

\bibitem{D} N.~Dutertre: Radial index and Poincar\'e-Hopf index of 1-forms on semi-analytic sets.
Math. Proc. Cambridge Philos. Soc. {\bf 148}, no.2, 297--330 (2010).

\bibitem{MMJ} W.~Ebeling, S.M.~Gusein-Zade: Indices of 1-forms on an isolated complete
intersection singularity. Mosc. Math. J. {\bf 3}, no.2, 439--455 (2003).

\bibitem{GD} W.~Ebeling, S.M.~Gusein-Zade: Radial index and Euler obstruction of a 1-form
on a singular variety. Geom. Dedicata {\bf 113}, 231--241 (2005).

\bibitem{MZ} W.~Ebeling, S.~M.~Gusein-Zade: Quadratic forms for a 1-form on an isolated complete intersection singularity. Math. Z. {\bf 252}, 755--766 (2006).

\bibitem{Sur} W.~Ebeling, S.M.~Gusein-Zade: Indices of vector fields and 1-forms on singular varieties. In: Global aspects of complex geometry, 129--169,
Springer, Berlin, 2006.

\bibitem{BLMS} W.~Ebeling, S.M.~Gusein-Zade:
Saito duality between Burnside rings for invertible polynomials.
Bull. Lond. Math. Soc. {\bf 44}, no.4, 814--822 (2012).

\bibitem{MMJ2} W.~Ebeling, S.M.~Gusein-Zade:
Orbifold Euler characteristics for dual invertible polynomials.
Mosc. Math. J. {\bf 12}, no.1, 49--54, 214 (2012). 

\bibitem{EGS} W.~Ebeling, S.M.~Gusein-Zade, J.~Seade:
Homological index for 1-forms and a Milnor number for isolated
singularities. Int. J. of Math. \textbf{15}, 895--905 (2004).

\bibitem{GLM} S.M.~Gusein-Zade, I.~Luengo, A.~Melle-Hern\'andez:
On an equivariant version of the zeta function of a transformation.
arXiv: 1203.3344.

\bibitem{G-MM} X.~G\'omez-Mont: An algebraic formula
for the index of a vector field on a hypersurface with an isolated
singularity. J. Alg. Geom. \textbf{7}, 731--752 (1998).

\bibitem{GSV} X.~G\'omez-Mont, J.~Seade, A.~Verjovsky:
The index of a holomorphic flow with an isolated singularity. Math. Ann.
\textbf{291}, 737--751 (1991).

\bibitem{GM} M.~Goresky, R.~MacPherson:
Stratified Morse theory. Springer-Verlag, Berlin, 1988.

\bibitem{Hall} M.~Hall Jr.: Combinatorial theory. Second edition.
Wiley-Interscience Series in Discrete Mathematics. A Wiley-Interscience Publication.
John Wiley \& Sons, Inc., New York, 1986.

\bibitem{HH} F.~Hirzebruch, Th.~H\"ofer:
On the Euler number of an orbifold.
Math. Ann. {\bf 286}, no.1-3, 255--260 (1990).

\bibitem{Luck} W.~L\"uck, J.~Rosenberg: The equivariant Lefschetz fixed point theorem
for proper cocompact $G$-manifolds. In: High-dimensional manifold topology, 322--361,
World Sci. Publ., River Edge, NJ, 2003.

\bibitem{MacPh} R.~MacPherson: Chern classes for
singular varieties. Annals of Math. \textbf{100}, 423--432 (1974).

\bibitem{MHSch} M.-H.~Schwartz: Classes caract\'eristiques
d\'efinies par une stratification d'une vari\'et\'e analytique complexe.
C. R. Acad. Sci. Paris S\'er. I Math. \textbf{260}, 3262--3264,
3535--3537 (1965)

\bibitem{SS} J.~A.~Seade, T.~Suwa: A residue formula for
the index of a holomorphic flow. Math. Ann. \textbf{304}, 621--634 (1996).

\bibitem{TtD} T.~tom Dieck: Transformation groups and representation theory.
Lecture Notes in Mathematics, 766, Springer, Berlin, 1979.

\bibitem{Verdier} J.-L.~Verdier: Caract\'eristique d'Euler-Poincar\'e. Bull. Soc. Math. France {\bf 101},
441--445 (1973).

\bibitem{Wall1} C.T.C.~Wall: A note on symmetry of singularities. Bull. London Math. Soc. {\bf 12},
no.3, 169--175 (1980).

\end{thebibliography}