Dear all, 

The  "Bernstein center" of a $p$-adic reductive group appears frequently in the literature of automorphic forms, often without a precise definition.   For example, in page 233 of Moeglin-Waldspurger's classic "Spectral decomposition and Eisenstein series" , the couple tell us : 

"...in particular the centre of enveloping algebra acts on $\delta$  via a character at the infinite places and the Bernstein centre does so at the finite places... "

So one may guess that it is some analogy of "the centre of enveloping algebra" at fintie places. 

My questions are:


What is the definition of the Bernstein centre of a p-adic reductive group.
What is the original motivation to introduce it ?
What role does it play in the theory of automorphic forms ?
Could you explain these in some concrete example ,say $GL_2$ ?


Please feel free to choose part of the questions to reply.
Any comments and references (in English) will also be very welcome !

Thank you very much in advance!

 Abstract Definition .
Let $Rep(G)$ be the abelian category of smooth complex representations of our $p$-adic group $G$. The Bernstein center is the endomorphism ring $\mathfrak Z(G)$ of the identity functor of $Rep(G)$. So it acts on any smooth representation, and this action commutes with any $G$-morphism.

 As a projective limit . Let $H$ be a compact open subgroup. Letting $\mathfrak Z(G)$ act on the permutation representation $\mathbb C[G/H]$ gives a morphim to the center $Z(G,H)$ of the Hecke algebra $\mathcal H(G,H)$. This yields an isomorphism $\mathfrak Z(G) \simeq \lim\limits_{\leftarrow H} Z(G,H)$ where transition maps are given by applying idempotents.

 Geometric realization. (here "geometric" is in the sense of Trace formulas, i.e. on the side of harmonic analysis). $\mathfrak Z(G)$ acts on the regular representations $C^\infty_c(G)$. The pairing $(z,f)\mapsto z.f(1)$ embeds $\mathfrak Z(G)$ as a set of distributions on $G$. The image is the convolution algebra of "essentially compact invariant distributions".

Spectral realization . By Schur's lemma (which holds in this context), $\mathfrak Z(G)$ acts on any irreducible representation $\pi$ via  a character $\theta_\pi:\mathfrak Z(G)\longrightarrow \mathbb C$. This is sometimes called the "infinitesimal character" of $\pi$, by analogy with the Archimedean situation, although there is nothing "infinitesimal" here. We get in this way a realization of $\mathfrak Z(G)$ as an algebra of continuous functions on the smooth dual $\hat G$ of $G$ equipped with the Fell topology. 

So far, nothing deep. Now, two major achievements in the representation theory of p-adic groups are 

 the Bernstein theorem which describes the spectral realization explicitly, 
the Harish Chandra Plancherel formula which provides a link between both realizations.


Let me try to describe Bernstein's result. Bernstein first splits the category $Rep(G)$ as a (infinite) product of indecomposable abelian subcategories (called "blocks"). Accordingly, the smooth dual decomposes into infinitely many connected components, and the center decomposes as an infinite product of rings.

The simplest example is that of a compact $G$ (e.g. the kernel of the norm map in a division algebra). In this case, $\hat G$ is discrete and the center is a product of copies of $\mathbb C$ indexed by the set of classes of irreps.
The next example is that of a compact-mod-center $G$ (e.g. the unit group of a division algebra). In this case there is an action of the group $\Psi(G)$ of unramified characters of $G$ on $\hat G$. Note that $\Psi(G)$ is naturally an algebraic torus over $\mathbb C$ because $G$ mod its maximal compact subgroup is a free abelian group of finite type. Now, connected components are the orbits of $\Psi(G)$, and the topology is the homogeneous space topology. In particular each such orbit carries a natural structure of an algebraic variety over $\mathbb C$. Finally, $\mathfrak Z(G)$ is the direct product of the ring of regular functions on these orbits.

Let us go to the general case. 
Assume first that $G$ is semisimple. Then each supercuspidal representation gives an isolated point in the smooth dual $\hat G$, because such representations are both projective and injective objects. Therefore, denoting by $Cusp(G)$ the set of (isom classes of) supercuspidal irreps, the ring $\mathbb C^{Cusp(G)}$ is a factor of $\mathfrak Z(G)$. If $G$ is reductive, then $Cusp(G)$ is still open and closed in $\hat G$, but is not discrete. As above, $\Psi(G)$ acts on $Cusp(G)$ and the latter is the disjoint union of orbits under $\Psi(G)$ with the natural quotient topology. The corresponding product of ring of regular functions on each orbits is then a factor of $\mathfrak Z(G)$, the "cuspidal" part $\mathfrak Z(G)_{cusp}$ of $\mathfrak Z(G)$.

Now remains the crucial step of describing the non-cuspidal part of the center. Bernstein uses parabolic induction from Levi subgroups. One quick way of stating the final result is :
$$ \mathfrak Z(G)\simeq \left( \prod_M \mathfrak Z(M)_{cusp} \right)^G$$
Here the product is over all Levi subgroup and $G$ acts by conjugacy. In order to get something less frightening, one can fix a maximal split torus $T$, restrict the product to those $M$'s that contain $T$ (finitely many) and take $N_G(T)$-invariants.

In particular, connected components are labeled by conjugacy classes of $\Psi(M)$-orbits
 in $Cusp(M)$. 
Suppose $G$ is split for simplicity. A particularly interesting component is that which corresponds to the $\Psi(T)$ orbit of the trivial character of $T$. Its contribution to 
$\mathfrak  Z(G)$ is isomorphic to $\mathbb C [X(T)]^W$. 
This component contains the unramified representations and the action of $\mathbb C[X(T)]^W$ on each such representation is given by its Satake parameter. In fact the corresponding connected component of $\hat G$ is the set of irreps that have non-trivial invariant under an Iwahori-subgroup, and one recovers the fact that the center of the Hecke-Iwahori algebra identifies with $\mathbb C[X(T)]^W$.
Finally, note that this ring looks like a "group version" of the center of an enveloping algebra, so the analogy with the Archimedean context is even deeper than expected.



Edit : answers to comments.

 On the link with Plancherel formula.  Denote by $\hat G_u$ the unitary smooth dual inside $\hat G$. The abstract Plancherel theorem says that there is a measure $d\pi$ on $\hat G_u$ such that for any $f \in C^\infty_c(G)$ (even for $f$ in the Schwartz algebra)  we have
$$ \int_{\hat G_u} Tr(\pi(f)) d\pi = f(1). $$
Note that $d\pi$ depends on the choice of a Haar measure on $G$ and the same choice is done to let $f$ act on $\pi$.
Let $z\in \mathfrak Z(G)$, applying the formula to $z.f$ yields the formula
$$ \int_{\hat G_u} Tr(\pi(f)) \theta_\pi(z) d\pi = (z.f)(1) $$
which expresses the distribution associated to $z$ (geometric realization) in terms of the
its action on $\hat G_u$.
Of course this is only useful if one has an explicit formula for the measure $d\pi$. Such a formula is provided by Harish Chandra's theorem. 

The first point is that the Plancherel measure is supported in the tempered spectrum.
The latter decomposes as a disjoint union of infinitely many connected components in a similar fashion to the full smooth dual, except that now one considers  $\Psi_u(M)$-orbits of discrete series, where $\Psi_u(M)$ now denotes unitary unramified characters (a compact torus inside $\Psi(M)$). So each component is a quotient of some homogeneous space under some $\Psi_u(M)$. The second point is that on such a tempered component, the Plancherel measure is absolutely continuous w.r.t. to the natural Lebesgue measure, and in fact given by some rational function. The precise computation of this rational function is not given by Harish Chandra, and in fact I don't know if it is known in general (maybe OK for classical groups).

 Relation to Galois representations  Assume here that $G=GL_n$ and consider the Langlands correspondence $\pi \mapsto \sigma(\pi)$ between irreducible representations of $G$ and Weil-Deligne representations. Then $\pi$ and $\pi'$ lie in the same Bernstein component if and only if the restriction to inertia of the Weil group action on $\sigma(\pi)$ and $\sigma(\pi')$ are isomorphic. I guess that they are tempered iff the monodromy filtration of $\sigma(\pi)$ is pure of weight $0$ and in this case $\pi$ and $\pi'$ are in the same tempered component iff restrictions of $\sigma(\pi)$ and $\sigma(\pi')$ to (Inertia $\times \langle N\rangle$) are isomorphic.

 Applications in global context  In the Langlands-Kottwitz approach to counting points on Shimura varieties in hyperspecial level, a certain spherical function (depending on the Shimura datum) has to be plugged in the Trace formula. In Iwahori level, work of Haines, Rapoport, Kottwitz shows that one should plug an element of the centre of the Iwahori-Hecke algebra. They then conjectured that similar phenomena should happen in deeper level.

More generally, recent work of Scholze shows (if I understood something, maybe he will contradict me soon!) that if one fixes an element $\tau$ in the Weil group (but outside the inertia group) and one considers the function
$\pi \mapsto Tr(\tau, \sigma(\pi))$, then one gets an element of $z_\tau\in \mathfrak Z(G)$ such that for $H$ a congruence subgroup, the trace of $\tau$ on the cohomology of a (suitable) Shimura variety of local level $H$ can be computed using a trace formula evaluated at a test function whose local factor is (associated to) the function $z_\tau.e_H$. 

I'm not able to provide more details on it, but the slogan is : central elements (should) provide good test functions in counting points on Shimura varieties.
