Homological duality for covering groups of reductive p-adic groups
Dragoş Frăţilă and Dipendra Prasad

To Dick Gross, with admiration.

In this largely expository paper we extend properties of the homological duality functor
where $\cH$ is the Hecke algebra of a
reductive $p$-adic group, to the case where
it is the Hecke algebra of a
finite central extension of a reductive $p$-adic group.
The most important properties being that $\RHom_\cH(-,\cH)$ is concentrated in a single degree for irreducible representations and that it gives rise to Schneider–Stuhler duality for Ext groups (a Serre functor like property).
Our simple proof is self-contained and bypasses the localization techniques of [31, 8] improving slightly on [25].
Along the way we also study Grothendieck–Serre duality with respect to the Bernstein center and provide a proof of the folklore result  that on admissible modules this functor is nothing else but the contragredient duality.
We single out a necessary and sufficient condition for when these three dualities agree on finite length modules in a given block.
In particular, we show this is the case for all cuspidal blocks as well as, due to a result of Roche [29], on all blocks with trivial stabilizer in the relative Weyl group.

§ INTRODUCTION

Let $G$ be a reductive $p$-adic group or a covering group (finite central extension) of such a group.
The homological duality for the abelian category  $\cM(G)$ of smooth representations of $G$
is defined at the level of derived categories as

\[ D_h:=\RHom_\cH(-,\cH)\colon \cD^b(\cM(G))\to \cD^b(\cM(G))^\op\]

where $\cH$ is the Hecke algebra of $G$.

An important property established by Bernstein in his Harvard notes [7] is that if $\pi$ is irreducible then $D_h(\pi)$ is concentrated in a single degree (and is irreducible).
We will denote this representation by $\bD_h(\pi)$.

On the other hand, in [31] Schneider and Stuhler also prove this property using localization techniques on the
Bruhat–Tits building.
Moreover, they show  that this functor $\bD_h$ enjoys a Serre functor like property, namely that for any irreducible representation $\pi$ of $G$ and any smooth representation $\pi'$ of $G$, there is a natural isomorphism

\begin{align}\label{Eq:intro Sch-Stuh duality}
\Ext^i_G(\pi,\pi')^*\simeq  \Ext_G^{d(\pi)-i}(\pi',\bD_h(\pi)^\vee),
\end{align}

where $d(\pi)\geq 0$ is an integer  which is bounded by the split rank of $G$ that we define later. Actually in [31], the isomorphism (<ref>) was proved only in the subcategory of representations with a fixed central character.
The more general version was proved in [25] and [4].

One of the aims of this work is to present a direct proof of (<ref>) for covering groups that does not make use of [31] or of localization techniques.
The strategy is to first show that the full homological duality $D_h$ satisfies a Serre functor like property

\[ \RHom_\cH(\pi,\pi')^*\simeq \RHom_\cH(\pi',D_h(\pi)^\vee) \]

for any smooth representations $\pi$ and $\pi'$ with $\pi$ finitely generated.
This is easy and follows from basic adjunctions naturally extended to idempotented algebras.
For convenience of the reader, and for lack of a precise reference, we provide all the details in <ref>.

The second step, namely showing that $D_h(\pi)$ is concentrated in a single degree for an irreducible representation, is non-formal and is based on several ingredients: Bernstein decomposition, second adjointness and a special property of the algebra governing a cuspidal block (Frobenius-symmetric algebra over its center).
Although the proof is already in [7] we hope that the argument that we present is more streamlined.

Another property of the homological functor $D_h$ is that on finite length cuspidal representations it agrees (up to a shift) with the contragredient functor.
A sketch of this result for irreducible representations appears already in [7] but we were not able to supply the details so we decided to include a different proof with full details.
To this effect we were led to study the Grothendieck–Serre duality over the Bernstein center $D_{GS/\cZ}$ (see <ref> for the definition).
In particular, we provide a proof of the folklore result that $D_{GS/\cZ}$
agrees with the contragredient for admissible representations.
Moreover we identify a necessary and sufficient condition (see condition (FsG) in Definition <ref>)
for the two functors $D_h$ and $D_{GS/\cZ}$ to agree (up to a shift) on a Bernstein block.
In order to state the condition, we need to introduce some notation.

If $\fs=[L,\rho]$ is a cuspidal datum, up to conjugation and inertia,  then there is an equivalence of the Bernstein block $\cM(G)_\fs$ with the module category $\rmod\cR_\fs$ for some algebra $\cR_\fs$ with center $\cZ_\fs$ that can be described explicitly (see <ref>).
The condition alluded to before, which we call Frobenius-symmetric-Gorenstein, since it is reminiscent of both these properties, is

\[ D_{GS/{\cZ_\fs}}(\cR_\fs)\simeq \cR_\fs[d] \text{ as $\cR_\fs$-bimodules}\]

where $d=d(\fs)$ is the split rank of the center of $L$ (equals the Krull dimension of $\cZ_\fs$).
We are able to check this condition on a cuspidal block $\fs=(\rho,G)$ because in this case $\cZ_\fs$ is a Laurent-polynomial algebra and $\cR_\fs$ is an  Azumaya algebra (see <ref> and <ref>).
Therefore we deduce that
the homological duality and the Grothendieck–Serre duality agree with the contragredient duality on all finite length cuspidal representations
in a given block.
Moreover, if $\fs=[L,\rho]$ is a cuspidal datum, by a result of Roche <cit.>, parabolic induction induces an equivalence of the blocks $\cM(L)_{[\rho]}\to\cM(G)_{\fs}$ if and only if the stabilizer of the inertia class $[\rho]$ in the relative Weyl group $N_G(L)/L$ is trivial, i.e., if and only if there is no non-trivial $w\in N_G(L)/L$ such that ${}^w\rho \simeq \rho\chi$ for some unramified character $\chi$ of $L$.
We also deduce that  for these blocks,  $D_h$ and $D_{GS/\cZ}$ agree (up to a shift).

There is yet another duality[It is indeed involutive but  covariant and  so the name duality is slightly misleading.] on smooth representations, namely the Aubert–Zelevinski duality.
Originally, it arose in representation theory of finite
groups of Lie type where it was defined on the Grothendieck group of finite dimensional representations.
Deligne–Lusztig introduced what is now called the Deligne–Lusztig complex in order to have a definition of the involution
at the level of representations.
For $p$-adic groups as well as their covering groups, the involution so constructed
is usually called the Aubert–Zelevinski involution.
It can be proved ([3, 4, 8]) that the Aubert–Zelevinski involution defines an exact functor
from the abelian category $\cM(G)$ to itself, hence it
extends trivially to a functor on the derived category $\cD^b(\cM(G))$:

\[D_{AZ}\colon \cD^b(\cM(G))\to \cD^b(\cM(G)).\]

The same involution was considered by Schneider–Stuhler in [31] and by Bezrukavnikov in <cit.> where the following isomorphism of functors was  proved using localization techniques

\begin{align}\label{intro:D_AZ D_GS=D_h}
D_{AZ}\circ D_{GS/\cZ}\simeq D_h.
\end{align}

Actually, in [31] this was shown only for finite length modules and in the Grothendieck group whereas in [8] the more general result was established.
The isomorphism (<ref>) was recently revisited in [4] where a simpler proof was given using the geometry of the wonderful compactification.

At the level of Grothendieck groups, it was  shown in [3] that $D_{AZ}$ commutes with the contragredient and the  commutation formulae with the parabolic induction and restriction functors were proved.
None of these are known for representations, say of finite length, of covering groups,
as even for linear groups, all these
assertions are proved by relating Aubert–Zelevinski involution to the homological duality (see (<ref>)) which is not yet
available for covering groups.

Results similar to Aubert's were proved in the context of smooth representations of covering groups in the work of Ban–Jantzen [12, 13] (again for the Grothendieck group of finite length representations).
In this work we do not address the question of whether (<ref>) holds for covering groups nor do we study the Aubert–Zelevinsky involution in this context.
An immediate obstacle to generalizing the approach of [4] is that as
covering groups are no longer linear,  we do not have at our disposal an obvious wonderful compactification.
However, in favor of the isomorphism (<ref>) is the fact that $[d]\circ D_h=D_{GS/\cZ}$ on cuspidal blocks, a result that we prove in <ref>.
By the aforementioned result of Roche, this continues to be the case for blocks $\fs=[L,\rho]$ such that the stabilizer of the inertia class of $\rho$ is trivial in the relative Weyl group $N_G(L)/L$.

One reason for interest in homological duality, and the attendant Serre functor property, is in the context of
Ext analogues of branching laws, as discussed in [27]. Recall that usually the branching laws in the context of $p$-adic group are considered for a representation
$\pi_1$ of a reductive group $G$  to a representation $\pi_2$ of a closed subgroup $H$, as $\Hom_H(\pi_1,\pi_2)$ (and never as
$\Hom_H(\pi_2,\pi_1)$). It has been suggested
in [27] that the success of these branching laws in the various cases studied stems from the vanishing of $\Ext^i_H(\pi_1,\pi_2), i>0$,
whereas by the Serre functor property,  $\Ext^i_H(\pi_2,\pi_1) = 0, i< d(\pi_2)$.
In particular,  $\Hom_H(\pi_2,\pi_1)$ is typically zero (so no wonder it is never considered!), and shows up only through the higher extension groups, i.e.,  $\Ext^{d(\pi_2)}_H(\pi_2,\pi_1)$.

A big chunk of the paper is expository: we give an exposition of some basic representation theoretic results in this context to show that they generalize in a  naive way to the case of covering groups culminating with Bernstein's decomposition (following closely the linear setting). We include a short discussion of Grothendieck–Serre duality limiting ourselves to the context that is sufficient for the applications we have in mind.
The reader familiar with the above classical results should directly consult <ref>, <ref>, <ref>.

Below follows a more precise description of our work.
Let $G=\bG(F)$ be a $p$-adic reductive group and let $\wG$ be a finite covering group of $G$.
Denote by $\cH$ the Hecke algebra of $\wG$ and by $\cM(\wG)$ the category of smooth complex representations of $\wG$.
The Levi subgroups and parabolic subgroups of $\wG$ are, by definition, the pullbacks of those of $G$.
We have parabolic induction and restriction functors (Jacquet modules)  defined as in the linear case.
A representation is called cuspidal if it is killed by all proper parabolic restriction functors.

We denote by $\cB(\wG)$ the equivalence classes (conjugation and inertia) of pairs $(\wL,\rho)$ of a Levi subgroup $\wL$  and an irreducible cuspidal representation $\rho$ of $\wL$.

The following is the Bernstein decomposition for $\wG$ (see [7, 6, 28] for the linear case):

We have a block decomposition

\[ \cM(\wG)\simeq \prod_{\fs\in \cB(\wG)} \cM(\wG)_\fs. \]

Moreover, each block $\cM(\wG)_\fs$ is equivalent to the category of modules over some $\bC$-algebra $\cR_\fs$ containing a finitely generated commutative subalgebra over which it is finite.

We will outline the proof in <ref> which follows closely the linear case as exposed in [7] or [28].
To that end, and to fix the notation, we will collect in Sections 1, 2, 3, 4 all the necessary results from the classical theory (i.e., the linear case) as well as some rudiments from category theory that go into the proof of it.
This is meant to convince the reader that the same strategy as in the linear case gives also the Bernstein decomposition for covering groups.

Using the second adjointness theorem of Bernstein, we prove the equivalence of
$\cM(\wG)_\fs$ with $\rmod\cR_\fs$ in <ref>.
For completeness, we also give in <ref>,  a skeleton of the proof of the second adjointness theorem
which is meant to convince the reader that the proof from the linear case (for which the first complete proof was
published by Bushnell in		[10])         goes through without changes for covering groups.
In doing this, we have followed the exposition in <cit.> of the linear case which makes use of completion functors in order to streamline the arguments.

The argument we present follows [28] rather than [7] in that it uses completion
of modules in order to streamline the proof.
We take advantage of this opportunity to introduce the completion functors in <ref> and we show that (at least on admissible representations) it allows one to recover the lost properties of the invariants $(-)^N$ functor.
See also <ref> for a far more advanced version (essentially equivalent to the second adjointness theorem).

For an irreducible representation $\pi$, the block $\fs=[\wL,\rho]$ that corresponds to $\pi$
is called  the cuspidal support of $\pi$. We denote by $d(\pi)=d(\fs)=d(\wL)$, the dimension of a maximal split torus in the center of the algebraic group $\bL$.

The category of (smooth) representations of $\wG$ has finite global dimension (see <ref>).
Consider the homological duality functor $D_h$ on the bounded derived category of $\wG$-modules

\[ D_h:=\RHom_\cH(-,\cH)\colon \cD^b(\cM(\wG))\to \cD^b(\cM(\wG))^\op. \]

It is surprisingly easy to show that[In the context of finite dimensional algebras, this functor is called Nakayama duality, and it was observed in <cit.> that it is a Serre functor.] $(\,)^\vee \circ D_h$ satisfies a Serre-functor-like property for the full $\RHom$, namely:
(see <ref>)
For $\pi, \pi' \in \cD^b(\cM(\wG))$ with $\pi$ finitely generated, we have a natural pairing of complexes of $\bC$-vector spaces

\[ \RHom_\cH(\pi,\pi')\otimes^L_\bC  \RHom_\cH(\pi',D_h(\pi)^\vee)\to \RHom_\cH(\pi,D_h(\pi)^\vee)\to \bC\]

providing a natural isomorphism

\[ \RHom_\cH(\pi,\pi')^* = \RHom_\cH(\pi',D_h(\pi)^\vee) \]

where $(-)^*$ is taking the dual vector space degree-wise.

Notice that we do not claim that $\RHom_\cH(\pi,D_h(\pi)^\vee)\to \bC$ is an isomorphism in general but merely that such a canonical map exists.

For a parabolic $\wP$ with Levi decomposition $\wP=\wL N$,  we denote by $\bfi_{\wL,\wP}^\wG$ (resp., $\bfr_{\wL,\wP}^\wG$) the parabolic induction (resp., restriction) functors.
Here are further homological properties that one can prove about $D_h$:

The functor $D_h\colon\cD^b(\cM(\wG))\to (\cD^b(\cM(\wG)))^\op$ enjoys the following properties

*  If $\pi\in\cM(\wG)_\fs^\finl$ is a finite length representation in a fixed Bernstein block, then $D_h(\pi)$ is concentrated in degree $d(\fs)$,

*  $D_h^2\simeq \Id$,

*  If $\pi$ is cuspidal of finite length and lives in a single block, then

\[D_h(\pi) = \pi^\vee[-d(\pi)],\]

* $D_h \circ \bfi_{\wL,\wP}^{\wG} = \bfi_{\wL,\wP^-}^{\wG} \circ D_h$ and  $D_h\circ \bfr_{\wL,\wP}^{\wG} = \bfr_{\wL,\wP}^{\wG} \circ D_h$,

*  $\bD_h:=D_h[d(\fs)]$ restricted to finite length representations in $\cM(\wG)_\fs$ is an exact involution providing an equivalence

\[ \bD_h\colon\cM(\wG)_\fs^{\finl}\xrightarrow{\sim} (\cM(\wG)_{\fs^\vee}^{\finl} )^\op,\]

where $\fs^\vee$ is the contragredient block of $\fs$

Part (<ref>) is proved in <ref>.
The involution statement is shown in <ref>.
Part (<ref>) is showed in <ref> after a quick interlude on the Grothendieck–Serre duality.
The commutation with parabolic induction and restriction is dealt with in <ref>.
The last part is a consequence of (<ref>),(<ref>) and (<ref>).
See also <ref>.

Let us say a few words about the Grothendieck–Serre duality which we will briefly
review in <ref> in the generality that we need.
Any representation $\pi\in\cM(\wG)$, besides having an action of $\wG$, has also an action of the Bernstein center $\cZ$, and by definition these two actions commute.
Therefore, any operation that one does to a $\cZ$-module will produce again a $\wG$-module.
The most trivial operation one can think of is taking the contragredient.
This works well, in the sense that it is an involution, on admissible representations.
One would like to extend the contragredient  in a functorial  way to all finitely generated modules such that it stays an involution.
Working with a fixed Bernstein block at a time, the question now is sent in the commutative algebra court: we have a nice ring (invariants under a finite group of a Laurent polynomial algebra, in particular Cohen–Macaulay) and finitely generated modules over it.
We would like to extend the contragredient from finite length modules to all finitely generated modules in such a way that it stays an involution.
One way to do this, coming from algebraic geometry, is to use the Grothendieck–Serre duality.
Although all the properties are nice and formal, one must now work in the derived category since the abelian category of modules is not preserved anymore under this duality.

For a commutative regular ring $A$ of Krull dimension $d$ (always containing a field $k$), the normalized dualizing (or canonical) complex is defined to be $\omega_A^\circ:=\omega_A[d]\in\cD^b(A\lmod)$, where $\omega_A=\Lambda^d\Omega_A^1$ is the module of top differentials on $A$.
If $A\to B$ is a finite map of commutative rings and $\omega_A^\circ$ is the normalized dualizing complex of $A$, then $\omega_B^\circ:=\RHom_A(B,\omega_A^\circ)$ is the normalized dualizing complex of $B$.

In our context, the (component of the) Bernstein center $\cZ_\fs$ is Cohen–Macaulay hence it is finite free over a regular subalgebra.
The above paragraph allows us to see that it has a normalized dualizing complex which moreover lives in a single degree (this is a characterization of Cohen–Macaulay rings).

For a commutative ring $A$ with a normalized dualizing complex $\omega_A^\circ$, one defines the Grothendieck–Serre duality as

\begin{align*}
D_{GS} &\colon \cD^b(A\lmod)\to (\cD^b(A\lmod))^\op \\
M & \mapsto \RHom_A(M,\omega_A^\circ).
\end{align*}

This has the nice property of being an involution $D_{GS}^2\simeq \Id$ when restricted to finitely generated modules. It is not obvious how to compute $D_{GS}(M)$ for a particular module $M$.
In general, it is a complex of modules even if $M$ is only a module.

However, a particular case in which  $D_{GS}(M)$  is easy to compute is that of finite
length modules $M$ (over an arbitrary ring $A$ containing the field $k$!).
Indeed, we have $D_{GS}(M)=M^*$ for any $M\in A\lmod$ of finite length.
For convenience of the reader we include (the short) proof of this in <ref>.

Going back to the category of smooth representations of $\wG$, we have the Grothendieck–Serre duality with respect to the Bernstein's center (component by component, see <ref>.)

\begin{align*}
D_{GS/\cZ}&\colon\cD^b(\cM(\wG)) \to (\cD^b(\cM(\wG)))^\op\\
\pi & \mapsto \RHom_\cZ(\pi,\omega_\cZ^\circ)^\sm
\end{align*}

where the superscript “$\sm$” means taking smooth vectors.
On finitely generated representations it is an involution, i.e.,  $D_{GS/\cZ}^2\simeq \Id$.
In order to make sure we extended the contragredient from admissible to all finitely generated,  one needs to check that $D_{GS/\cZ}(\pi)=\pi^\vee$
for all admissible representations $\pi$ of $G$.
This is a folklore result whose (again easy and rather formal) proof we include for the convenience of the reader and for lack of a better reference (see <ref>).

However, beyond finite length representations, $D_{GS/\cZ}(\pi)$ seems not to have been
considered.  As a specific example,
we may mention that we do not know  what $D_{GS}(M)$ is for
$M = \ind_{K}^{G} \bC$, the space of locally constant
compactly supported $\bC$-valued functions on $K\backslash G$ for $K$ a compact open subgroup of $G$. Here is a more specific case. Suppose  $G$ is a split group, and $K=\I$ an Iwahori subgroup of $G$, with
$\HH(\I)=C_c(\I\backslash G/\I)$, the Iwahori Hecke algebra of $G$ and $Z$ the center of $\HH(\I)$. This
particular Bernstein component of $G$ (called the Iwahori block, consisting of smooth representations of $G$
generated by their $\I$-fixed vectors)
is described as the category of modules over $\HH(\I)$ by sending a smooth
representation $M$ of $G$
(generated by its $\I$-fixed vectors) to $M^\I$ which is a module for $\HH(\I)$.
Then for $M = \ind_{\I}^{G} \bC$, $M^\I = \HH(\I)$ with the left regular representation of
$\HH(\I)$ on itself. In this case,
is the $\HH(\I)$-module corresponding to $\Hom_Z(\HH(\I), Z)$, i.e.
$\Ext_Z^i(\HH(\I),Z)=0$ for $i>0$.
This follows first by noting that
<cit.>, $\ind_{\I}^{G} \bC$ is isomorphic to the universal unramified principal series $\ind_{B}^G(\Pi)$ where $\Pi= \ind_{T^o}^T \bC$, $B$ is a Borel subgroup of $G$, $T \subset B$ a maximal torus of $G$ and
$T^o$, the maximal compact subgroup of $T$. In this case, it follows from
<cit.> that $M^\I = \HH(\I)$ is a free module over $Z$,
proving that $\Ext_Z^i(\HH(\I),Z)=0$ for $i>0$. However, we do not know what $\HH(\I)$-module we get
for $\Hom_Z(\HH(\I),Z)$
— and we do not have any guesses to offer except to ask if $\Hom_Z(\HH(\I),Z) \cong \HH(\I)$? The corresponding question for
the group asks about  the structure of the $G$-module
$D_{GS}(M)$ for $M = \ind_{\I}^{G} \bC$.

Homological duality allows us to prove part (<ref>) of <ref>.
Additionally, it permits us to understand precisely under what conditions do we have an isomorphism $[d]\circ D_h\simeq D_{GS/\cZ}$ on a given block.
For precise details, we refer to <ref> and the condition (FsG) in Definition <ref>.

Returning to homological duality and its Serre functor property, let us state the main theorem proved
in this work.
For $\pi\in\cM(\wG)_\fs^\finl$, a finite length representation in a Bernstein block,
we put $\bD_h(\pi) = H^{d(\pi)}(D_h(\pi))$ to be the unique non-zero cohomology of $D_h(\pi)$ (see <ref>(<ref>)).

(see <ref>)
Let $\pi\in \cM(\wG)_\fs$ be of finite length and $\pi'\in \cM(\wG)$ be arbitrary.
Then the following natural pairing is perfect

\[ \Ext^i_{\wG}(\pi,\pi')\otimes \Ext^{d(\pi)-i}_{\wG}(\pi',\bD_h(\pi)^\vee)\to \Ext^{d(\pi)}_{\wG}(\pi,\bD_h(\pi)^\vee)\to \bC.\]

The proof is an easy consequence of <ref> and <ref>.
It is an improvement on the results of [25] where the theorem was proved only for $\pi$ irreducible.

* As in the linear case, it would be desirable to have an identification of $D_h$ with $D_{AZ}\circ (-)^\vee$ on the category of finite length representations of a covering group
$\wG$, where $D_{AZ}$ is the Aubert–Zelevinsky duality, similar to [4], <cit.> (Grothendieck group) or <cit.>.
For the moment, it seems we do not know that $D_h(\pi) \cong D_{AZ}(\pi^\vee)$ even for an irreducible representation $\pi$ of $\wG$. We consider this as an important question
which is not at all discussed in this paper.
However, one can consult [13] for a discussion of Aubert–Zelevinsky duality for finite central extensions of reductive $p$-adic groups (in the Grothendieck group of representations).

* As already mentioned, the duality <ref> was first proved in <cit.> using a more involved argument.
Their strategy was to first show the vanishing of Ext groups from
<ref>  (<ref>) and then proceed to prove the isomorphism from <ref> using this vanishing.
Both steps hinge on nice projective resolutions constructed from the Bruhat-Tits building.
The advantage of the proof that we present is that by keeping the homological duality functor $D_h$ at the derived level, the duality theorem becomes very easy and requires no technology.
Once the vanishing of Ext-groups <ref>(<ref>) is proved (for which we follow the argument in [7]) we immediately deduce the required duality theorem at the level of abelian categories.

D.F. would like to thank IIT Mumbai, where this work has started,  for their hospitality.
The second author  thanks  SERB, India for its support
through the JC Bose
Fellowship, JBR/2020/000006.
His work was also supported by a grant of
the Government of the Russian Federation
for the state support of scientific research carried out
under the  agreement 14.W03.31.0030 dated 15.02.2018.
We would like to thank Roman Bezrukavnikov for some very useful remarks concerning homological duality and Grothendieck–Serre duality. The authors
are grateful to the referees for a careful reading and helpful remarks.

§ CATEGORICAL GENERALITIES

§.§ Preliminaries on decomposing categories and centers

Given $I$ a set (possibly infinite) and $\cC_i, i\in I$, a family of abelian categories, we can define the product category $\cC:=\prod_{i\in I}\cC_i$ in the 2-category of abelian categories through the usual universal property.
Concretely, one constructs it as follows:

* the objects are tuples $(X_i)_{i\in I}$ with $X_i\in \cC_i$ for every $i\in I$;

* the morphisms are defined by

\[ \Hom_\cC((X_i)_i, (Y_j)_j):=\prod_{i\in I} \Hom_{\cC_i}(X_i,Y_i); \]

* the projection functors $\pi_i\colon \cC\to \cC_i$ send a tuple $(X_s)_s$ to $X_i$.

The universal property of a product is immediately verified.

In addition to the projection functors $\pi_i\colon\cC \to \cC_i$, there are also natural inclusion functors

\[ \iota_i\colon \cC_i\to \cC \]

sending an object $X\in \cC_i$ to the tuple that has $X$ in position $i$ and $0$ everywhere else.
By construction, the functor $\iota_i$ is fully faithful so we can think of the category $\cC_i$ as a full subcategory of the product category $\cC$.
We will drop the functor $\iota_i$ from the notation most of the time.
Two such subcategories $\cC_i$ and $\cC_j$ for $i\neq j$ are (derived) orthogonal, namely

\[ \Ext^k_\cC(X_i,X_j) = 0,\text{ for all }k\ge 0.\]

Moreover we notice that the object $(X_i)_i\in\cC$ is the direct sum of the objects $\iota_i(X_i)$ in $\cC$.
Indeed, let us check the universal property for direct sums: for an arbitrary object $Y=(Y_i)_i\in\cC$ we have

\begin{align*}
\Hom_{\cC}((X_i)_i, (Y_i)_i)& = \prod_i \Hom_{\cC_i}(X_i,Y_i)\\
& = \prod_i \Hom_{\cC}(\iota_i(X_i),(Y_j)_j)\\
& = \Hom_{\cC}(\oplus_i \iota_i(X_i),(Y_j)_j).
\end{align*}

Actually, the same argument shows that $(X_i)_i$ is also the direct product of the objects $\iota_i(X_i)$, $i\in I$, in $\cC$.

Caution: the above does not mean that in $\cC$  we always have $\oplus_i A_i\simeq \prod_i A_i$ for any $A_i\in\cC$.
See also <ref>.

As a summary, we have a category $\cC$ with full subcategories $\cC_i$, $i\in I$ that are two by two derived orthogonal and moreover every object of $\cC$ is a direct sum of objects from $\cC_i$ (the subcategories $\cC_i$ split the category $\cC$).
Conversely, a category $\cC$ with full subcategories $\cC_i$, $i \in I$, with the above properties is a direct product of
provided we assume a small technical condition on the categories that we work with which in practice is always verified.

(cf. <cit.>)
Let $\cC$ be an abelian category and let $\cC_i\subset \cC$, $i\in I$ be full abelian subcategories of $\cC$.
Assume that

*  $\cC$ admits direct sums indexed by $I$,

*  every object of $\cC$ can be written as a direct sum of objects of $\cC_i$,

*  $\Hom_\cC(X_i,X_j)=0$ for all $X_i\in\cC_i$, $X_j\in\cC_j$ and $i\neq j$.

*  If $X_i\in\cC_i$, $i\in I$ and $f\colon Y\to \oplus_i X_i$ is such that $\proj_i\circ f=0$ for all $i\in I$ then $f=0$.[There are pathological examples where this condition is not satisfied.]

Then the natural functor $\prod_i \cC_i \to \cC$ is an equivalence of categories.

There is a natural functor $\prod_i \cC_i \to \cC$ given by $(X_i)_i\mapsto \oplus_i X_i$ that is well defined by assumption (<ref>).
By (<ref>) it is essentially surjective.

In order to show that it is an equivalence it remains to check that it is fully faithful.

Given this observation and taking into account assumptions (<ref>) and (<ref>), we have

\begin{align*}
\Hom_{\prod_i \cC_i}((X_i)_i,(Y_i)_i)  & = \prod_i \Hom_{\cC_i}(X_i,Y_i)\\
& = \prod_i \Hom_{\cC}(X_i,\oplus_j Y_j)\\
& = \Hom_\cC (\oplus_i X_i,\oplus_j Y_j)
\end{align*}

which proves the fully-faithfulness.

The following example involving direct sum and direct product might be clarifying.

Consider the algebra $A=\bigoplus_{i\in \bZ} \bC = Func_c(\bZ,\bC)$, i.e., complex valued functions with compact support defined on $\bZ$ with pointwise multiplication.
It is an algebra that has no unit but it has enough idempotents.
We can consider the category of non-degenerate modules over $A$, call it $\cC:=A\lmod\nd$, where an object of $\cC$ is an $A$-module $M$ such that for any $m\in M$ there exists an idempotent $e\in A$ such that $em=m$.
By considering the delta functions $\delta_i$, $i\in\bZ$, it is clear that any non-degenerate module $M$ satisfies

\[M=\bigoplus_i M_i,\text{ where }M_i:=\delta_iM.\]

Notice also that if $M=M_i$ and $M'=M'_i$ then $\Hom_\cC(M,M') = \Hom_{\mathrm{Vec}_\bC}(M_i,M'_i)$.
Given two non-degenerate $A$-modules $M$ and $M'$ we see therefore that

\[\Hom_\cC(M,M') = \Hom_\cC(\oplus_i M_i,\oplus_j M'_j) = \prod_i \Hom_{\mathrm{Vec}_\bC}(M_i,M'_i)\]

because there are no $A$-module morphisms between $M_i$ and $M'_j$ if $i\neq j$.

For any $i\in\bZ$ we can consider the full subcategory of $\cC$ consisting of those modules $M$ on which $\delta_i$ acts as identity. In other words, modules $M$ such that $M=\delta_iM$. We denote it by $\cC_i$ and by what we said above we have $\cC_i\simeq\mathrm{Vec}_\bC$ for all $i\in \bZ$.
From the above Proposition we can deduce an equivalence of categories

\[\cC\simeq \prod_{i\in\bZ}\mathrm{Vec}_\bC.\]

Since all the categories $\cC_i$ are equivalent to $\mathrm{Vec}_\bC$ one could confuse the direct sum $\oplus_i^\cC M_i\in \cC$ where each $M_i$ is a vector space, to the direct sum $\oplus_i^{\mathrm{Vect}_\bC}M_i\in \cC$ where the superscripts refer to the category where the direct sums/products are taken. The former is an "external" direct sum whereas the latter is an "internal" direct sum (all the objects are seen as belonging to the same $\mathrm{Vec}_\bC$).
Moreover, we actually have $\oplus_i^\cC M_i \simeq \prod_i^\cC M_i$. One direct way to see this is to remember that $\cC = A\lmod\nd$ and that the direct product of non-degenerate modules is defined by

\begin{align*}
\prod_i{}^{A\lmod\nd}
M_i &=
\{(m_i)\in\prod_i{}^{\mathrm{Vect}_\bC} M_i
\mid \exists e\text{ idempotent in }A
\text{ such that } em=m\} \\
&= \{(m_i)\in\prod_i{}^{\mathrm{Vect}_\bC} M_i\mid \text{ only finitely many } m_i \text{ are non-zero}\}.
\end{align*}

Given an abelian category $\cC$, we define its center $\cZ(\cC)$ to be $\End(\Id_{\cC})$, the endomorphisms of the identity functor.

The center of a category is preserved under categorical equivalence.

Since $\cC$ is additive we see that $\cZ(\cC)$ is a commutative ring with unit.
Moreover, the category $\cC$ becomes naturally $\cZ(\cC)$-enriched, i.e., the $\Hom$ spaces in $\cC$ have a natural action of $\cZ(\cC)$ making them $\cZ(\cC)$-modules and $\cC$ into a $\cZ(\cC)$-linear category.

If $\cC = A\lmod$ for an algebra $A$ with unit, then it is not hard to see that $\cZ(\cC)$ is the center $\cZ(A)$ of $A$.

For $e\in \cZ(\cC)$ an idempotent we denote by $e\cC$ its image in $\cC$: it is the full subcategory consisting of those objects on which $e$ acts by identity.

A similar argument as in <ref> proves the first part of

If $\cC\simeq \prod_i \cC_i$ then $\cZ(\cC) \simeq \prod_i \cZ(\cC_i)$.
Conversely, if $\cZ(\cC) \simeq \prod_i Z_i$ then the identity of each $Z_i$ provides an idempotent $e_i\in \cZ(\cC)$ and  we have $\cC\simeq \prod_i e_i\cC$.

Only the last part is non-trivial and it follows by applying <ref>.

The following is a generalization of <ref> for algebras without unit but with enough idempotents.
An algebra $A$ is said to have enough idempotents if for any element $a\in A$ there exists an idempotent $e\in A$ such that $ae=ea=a$.

A left $A$-module $M$ is said to be non-degenerate if for any $m\in M$ there exists an idempotent $e\in A$ such that $em=m$.
The category of non-degenerate left $A$-modules is denoted as $A\lmod\nd$.

If $A^o$ is the opposite algebra to $A$,  then $\overline{A}:=\Hom_{A^o}(A,A)$ is the space of right $A$-invariant maps from $A$ to $A$.
We can write $\overline{A}= \Hom_{A^0}(\varinjlim eA,A)= \varprojlim Ae$ where the limit is taken over all the idempotents of $A$.
Clearly $\overline{A}$ is an $A$-bimodule.

The center of the category $A\lmod\nd$ is identified with the center of $\overline{A}$ as an $A$-bimodule (i.e., with the $A$-bimodule endomorphisms of $A$).
It can be further identified with $\varprojlim Z(eAe)$ where $e$ goes over all idempotents of $A$.

In our context, the abelian categories that we will encounter will be equivalent to module categories over a unitary ring and as such their center is just the center of the corresponding ring.

We use a standard tool from category theory to detect when an abelian category is equivalent to a module category.

Recall that a functor $F\colon \cC\to\cD$ is called faithful (resp., fully-faithful) if for any objects $A,B\in\cC$ it induces an injective (resp., bijective) map on Hom spaces:

\[ \Hom_\cC(A,B)\to \Hom_\cD(F(A),F(B)).\]

One calls $F$ essentially surjective if every object in $\cD$ is isomorphic to the image through $F$ of an object from $\cC$.

A basic theorem in category theory (see <cit.>) says that a functor $F\colon \cC\to \cD$ is an equivalence of categories if and only if it is essentially surjective and fully-faithful.

An object $P$ of an abelian category $\cA$ is called

* compact (or finite) if $\Hom_\cA(P,-)$ commutes with arbitrary coproducts(=direct sums),

* projective if $\Hom_\cA(P,-)$ is exact,

* a generator if $\Hom_\cA(P,-)$ is faithful, a notion that  we just recalled above
and which for $P$ a projective object amounts to the assertion that for $X \not = 0$,
$\Hom_\cA(P,X) \not = 0$.

* progenerator if it is projective and a generator.

Let $\cA$ be an abelian category admitting coproducts.
For a compact progenerator $P$ of $\cA$,  the functor

\begin{align*}
\cA & \to \rmod\End_\cA(P)\\
X&\mapsto \Hom_\cA(P,X)
\end{align*}

is an equivalence of categories between $\cA$ and the category of left modules over the ring $\End_\cA(P)$.

§.§ Serre functors

The notion of a Serre functor for an additive category was introduced in [14] in order to capture duality phenomena similar to Serre's duality theorem on smooth projective varieties.
It is intimately related to questions of representability of certain functors.
Moreover this turns out to be a very rigid notion.
In particular, if a Serre functor exists then it is unique (up to natural isomorphism).

Let $\cC$ be a $k$-linear category.
A Serre functor $\bS$ on $\cC$ is an equivalence of categories $\bS\colon \cC\to \cC$ together with natural isomorphisms

\[\eta_{A,B}\colon  \Hom_\cC(A,B)^*\to \Hom_\cC(B,\bS(A)), \,\text{ for all } A,B\in\cC.\]

It is observed in <cit.> that if one assumes that the categories $\cC_1,\cC_2$,  with Serre functors $\bS_1,\bS_2$,
have finite dimensional Hom spaces,  then any functorial isomorphism  between
the categories  $\cC_1,\cC_2$ commutes with the Serre functors.

An example of a Serre functor, actually one that motivated the notion, is the classical Serre duality.
Namely, let $X$ be a smooth projective variety over $k$ of dimension $d$ and denote by $\omega_X$ its canonical sheaf.
Then Serre duality stipulates a natural isomorphism

\[ \Ext^i_{\cO_X}(\cF,\cG)^* \simeq \Ext^{d-i}_{\cO_X}(\cG,\cF\otimes\omega_X), \text{ for all } \cF,\cG\in\cD^b(\Coh(X)).\]

In other words, the functor

\[ -\otimes \omega_X[d]\colon \cD^b(\Coh(X))\to \cD^b(\Coh(X)) \]

is a Serre functor.

A simpler example comes from finite dimensional algebras over a field $k$ and is known as the Nakayama functor, see <cit.>.
Let $A$ be a finite dimensional algebra over $k$ and suppose it has finite homological dimension.
(Although there are no nontrivial finite dimensional commutative algebras of finite homological dimension over a field $k$, there are many non-commutative ones, for example, there are  constructions of such algebras using Quivers.)
We consider the derived category $\cC:=\cD^b_{fd}(A\lmod)$ of finite dimensional left $A$-modules.
There are two duality functors

\[ \cD^b_{fd}(A\lmod) \stackrel{\delta}{\to} \cD^b_{fd}(\rmod A) \stackrel{(-)^\vee}\to \cD^b_{fd}(A\lmod),\]

where $\delta(M) = \RHom_A(M,A)$ and $M^\vee = \RHom_{\cD^b(k)}(M,k)$ for any object $M\in \cD^b(A\lmod)$.
We put $D_{Nak}:= (-)^\vee\circ \delta$ and call it the Nakayama functor.

The following proposition is very easy to prove from classical adjunctions  and our results in <ref> are essentially a detailed version of it for idempotented algebras (see <ref>):

The Nakayama functor $D_{Nak}\colon \cD^b_{fd}(A\lmod)\to\cD^b_{fd}(A\lmod)$ is a Serre functor.

Notice that we have  to restrict ourselves to finite dimensional modules because of  the appearance of the dual vector space which provides an equivalence of categories only for finite dimensional modules.

In order to extend the notion to more general categories, in particular to finitely generated modules over an algebra finite over its center, Bezrukavnikov and Kaledin propose the notion of relative Serre functor (see <cit.>).
Since we do not prove anything about relative Serre functors we prefer to defer this discussion to a later work.

§.§ An abstract duality theorem

The purpose of this section is to prove an abstract duality theorem reminiscent of the Serre functor property of the Nakayama functor $\RHom_A(-,A)^*$ (see [14] and <ref>).
This is achieved in <ref> and <ref>.
Although the results presented in this section are well-known and we do not claim any originality, for lack of a
precise reference,  we provide all the details. We do not strive for maximal generality, so sometimes we make hypotheses which might not be necessary but which hold in our applications to $p$-adic groups.

Let $k$ be a field and let $A$ be an idempotented $k$-algebra, i.e.,  for every $a\in A$ there exists an idempotent $e\in A$ such that $ae=ea=a$.
Clearly any unitary algebra is idempotented.
We suppose moreover that $A$ has a countable filtered set of idempotents.
A left $A$-module $M$ is said to be non-degenerate if $AM=M$, equivalently, if
for any $m\in M$, there exists an idempotent $e\in A$ such that $em=m$.

For a non-unitary ring $A$, a free $A$-module is not necessarily projective.
The basic projective left $A$-modules are $Ae$ with $e$ an idempotent.
Any projective finitely generated non-degenerate $A$-module is a direct summand of $\oplus_i Ae_i$ where $\{e_i\}$ is a finite collection of idempotents.
This follows as in the unitary case using the fact that the module is non-degenerate and finitely generated.

We denote by $A\lmod$, the category of all left $A$-modules, and by $A\lmod\nd$ the full subcategory of non-degenerate left $A$-modules.
We use a similar notation for right $A$-modules.

Consider the functors

\begin{align}\label{Eq:functors Hom and tensor product}
\begin{split}
\Hom_A(-,-)\colon &(A\lmod\nd)^{op}\times A\lmod\nd \to \bZ\lmod\\
-\otimes_A- \colon & \rmodnd A\times A\lmod \to \bZ\lmod
\end{split}
\end{align}

The category of non-degenerate left $A$-modules, $A\lmod\nd$, has enough projective objects[We do not know if the category of all left modules over $A$ has enough projective objects, but this plays no role for us.] and so we can  derive the functors (<ref>) on the first argument since non-degenerate projective modules are still acyclic:

\begin{align}\label{Eq:derived functors Hom and tensor product}
\begin{split}
\RHom_A(-,-)\colon &(\cD^b(A\lmod\nd))^{op}\times \cD^b(\rmodnd A) \to \cD(\bZ\lmod)\\
-\otimes^L_A- \colon & \cD^b(\rmodnd A) \times \cD^b(A\lmod) \to \cD(\bZ\lmod)
\end{split}
\end{align}

The functor $\Hom_A(-,A)\colon (A\lmod\nd)^{op}\to \rmod A$ does not land inside the subcategory of non-degenerate modules.
This can already be seen for $A$ itself: $\Hom_A(A,A) \simeq \varprojlim_e eA$ where the  inverse limit is taken over the poset of idempotents $e$ in $A$ (can take a filtered subset).
However, if $M$ is a finitely generated non-degenerate module then $\Hom_A(M,A)$ is non-degenerate.

For two $A$-modules $M,N\in A\lmod\nd$, there is a canonical morphism

\begin{align}\label{Eq:dual Hom to Hom natural morphism}
\can_{N,M}\colon \Hom_A(M,A)\otimes_A N\to \Hom_A(M,N)
\end{align}

that extends to the derived category $\cD^b(A\lmod\nd)$

\begin{align}\label{Eq:can Hom in derived cat}
\can_{M,M}\colon \RHom_A(M,A)\otimes_A^L N\to \RHom_A(M,N).
\end{align}

A non-degenerate module $M$ over $A$ is said to be perfect if it has a finite resolution by finitely generated non-degenerate projective $A$-modules.
An object of $\cD^b(A\lmod\nd)$ is said to be perfect if it is isomorphic to a finite complex of finitely generated non-degenerate projective $A$-modules.

The next lemma tells us that $\can_{N,M}$ is an isomorphism when $M$ is perfect:

If $M,N\in\cD^b(A\lmod\nd)$ and $M$ is perfect, the canonical morphism (<ref>)

\[ \RHom_A(M,A)\otimes_A^L N\to \RHom_A(M,N) \]

is an isomorphism.

In this proof all modules are non-degenerate.
We split the proof into several steps.

First notice that for $M=Ae$ with $e$ an idempotent we have canonical identifications $\RHom_A(Ae,N)=\Hom_A(Ae,N)=eN$ and in particular $\Hom_A(Ae,A)=eA$ is a projective right $A$-module and so one can compute the derived tensor product with it.
Hence in this situation we have

\[ \RHom_A(Ae,A)\otimes^L N = eA\otimes_A N=eN=\Hom_A(Ae,N) \]

and we are done.

Second, notice that the  morphism $\RHom_A(M,A)\otimes_A^L N\to \RHom_A(M,N)$
in the statement of the lemma
is compatible with finite direct sums (in both arguments).
Since a finitely generated projective module is a direct summand of $\oplus_{i=1}^n Ae_i$ for some idempotents $e_i$ we deduce the validity of the lemma for $M$ a finitely generated projective module.

Next, let $M$ be a perfect complex, quasi-isomorphic to $P_\bullet$ where each $P_r$ is a finitely generated non-degenerate projective module.

For $N=N_\bullet\in\cD^b(A\lmod\nd)$, $\RHom_A(M,N)$ is computed by the totalization of a double complex with entries $\Hom_A(P_r,N_s)$ which by what we said above is isomorphic to $\Hom_A(P_r,A)\otimes_A N_s$ through $\can_{P_r,N_s}$.
The latter are the entries of the double complex $\Hom_A(P_\bullet,A)\otimes_A N_\bullet$ whose totalization computes $\RHom_A(M,A)\otimes^L_A N$.
The naturality of (<ref>) ensures that these isomorphisms commute with all the differentials in the double complexes and therefore the total complexes are isomorphic.
In other words, the natural map $\can_{N,M}\colon \RHom_A(M,A)\otimes^L_A N\to \RHom_A(M,N)$ is an isomorphism.

For an $A$-module $P$, we denote by $P\nd$ its non-degenerate submodule, i.e.,
the subspace  of $P$ consisting of elements that are fixed by some idempotent in $A$.

It is clear that a morphism from a non-degenerate $A$-module to an arbitrary $A$-module lands inside its non-degenerate part.
Said otherwise, we have

The right adjoint of the natural inclusion $A\lmod\nd\to A\lmod$ is the functor $(-)\nd$ of taking the non-degenerate part of a module.
Moreover $(-)\nd$ is exact,  so the adjunction extends trivially to derived categories.

Contragredient. Any non-degenerate left or right $A$-module $M$ has a natural structure of $k$-vector space.
We can therefore construct the contragredient module $M^\vee:=\Hom_k(M,k)^{\nd}$ as the non-degenerate $k$-linear maps from $M$ to $k$.
As $M\rightarrow M^\vee$ is an exact functor from $A\lmod\nd$ to $\rmodnd A$, it extends naturally  to the bounded derived categories

\[ (-)^\vee\colon \cD^b(A\lmod\nd)\to\cD^b(\rmodnd A)  \]

by applying it degree-wise.

For a right $A$-module $M$, a left $A$-module $N$, both non-degenerate and a $k$-vector space $V$, we have a natural adjunction morphism

\[ \tau_{M,N,V}\colon \Hom_k(M\otimes_A N,V)\to \Hom_A(N,\Hom_k(M,V)^{\nd}) \]

that can be checked (as in the classical situation) to be an isomorphism.
This isomorphism  extends to bounded derived categories and we get the usual derived tensor-hom adjunction

\begin{align}\label{Eq:tensor hom adjunction non unitary}
\tau_{M,N,V}\colon \RHom_k(M\otimes_A^L N,V) \stackrel{\sim}{\to} \RHom_A(N,\RHom_k(M,V)^{\nd})
\end{align}

For a complex of vector spaces $V\in \cD^b(k)$, we denote by $V^*$, the complex of dual vector spaces.

Denote by $\cD^b_{\perf}(A\lmod\nd)$, the full subcategory of $\cD^b(A\lmod\nd)$ consisting of perfect complexes.

The following functor

\begin{align*}
D_{\Nak}\colon  \cD^b_{\perf}(A\lmod\nd)  & \longrightarrow  \cD^b(A\lmod\nd) \\
M\quad   & \mapsto  \RHom_A(M,A)^\vee,
\end{align*}

satisfies the Serre functor duality, i.e., for any perfect object $M$ and any object $N\in \cD^b(A\lmod\nd)$, we have a natural isomorphism of $k$-vector spaces:

\[ \Hom_{\cD^b(A)}(M,N)^* \simeq \Hom_{\cD^b(A)}(N,D_\Nak(M)). \]

A Serre functor (see <ref>) is,
moreover, required to be an equivalence of categories. In our situation, in order for $D_{\Nak}$ to be an equivalence, we must restrict to finite dimensional modules (because of duality over $k$).
In the context of representations of $p$-adic groups this means restricting to admissible modules.
Replacing the contragredient by Grothendieck–Serre duality over the Bernstein center extends $D_{\Nak}$ from admissible modules to finitely generated modules and indeed gives a Serre functor (relative to the center) for the whole category of finitely generated representations.[We are indebted to Roman Bezrukavnikov for explaining this to us.]
This statement already appears in [4].

Apply <ref> and the tensor-hom adjunction (<ref>) to get natural quasi-isomorphisms

\begin{align*}
\RHom_k( \RHom_{A}(M,N), k) & \simeq    \RHom_k(\RHom_A(M,A) \stackrel{L}{\otimes} _A N, k) \\
& \simeq    \RHom_A(N, \RHom_k(\RHom_A(M,A), k)^{\nd})\\
& \simeq \RHom_A(N,\RHom_A(M,A)^\vee)\\
& \simeq \RHom_A(N,D_\Nak(M)).
\end{align*}

Taking $\rH^0$ gives the desired result.

Putting $N=M, D_\Nak(M)$ above, we get the following:

Suppose $M$ is a non-degenerate irreducible left $A$-module which belongs to $\cD^b_\perf(A\lmod\nd)$ and for which Schur's lemma holds.
\[\Hom_{\cD^b(A)}(D_\Nak(M),D_\Nak(M) )^*\simeq k.\]

It follows from the following chain of natural isomorphisms

\[k\simeq \Hom_{\cD^b(A)}(M,M)^* \simeq \Hom_{\cD^b(A)}(M,D_\Nak(M))
\simeq\Hom_{\cD^b(A)}(D_\Nak(M),D_\Nak(M) )^*.\]

The above corollary shows that for $M$ irreducible, $D_\Nak(M)$ is indecomposable.
So if $D_\Nak(M)$ is concentrated in degree $0$, then it is isomorphic to $M$.
In particular, if $M$ is irreducible and projective, then $D_\Nak(M)\simeq M$.
If we apply it to a finite dimensional semisimple algebra $A$, then we get the simple fact that $D_\Nak$ leaves stable every irreducible module.
See <ref> for a statement for $p$-adic groups.
In general however, it looks like $D_\Nak(M)$ might be an interesting, non-trivial, involution.

Taking $N=D_\Nak(M)$ in the above theorem we get a canonical map

\begin{align}
\can_M\colon \Hom_{\cD^b(A)}(M,D_\Nak(M))\to k
\end{align}

and moreover, unwinding the definitions (see the proof of <ref>), we obtain
For $M,N\in\cD^b(A\lmod\nd)$ with $M$ perfect, the natural pairing

\[ \Hom_{\cD^b(A)}(M,N)\times \Hom_{\cD^b(A)}(N,D_\Nak(M))\to \Hom_{\cD^b(A)}(M,D_\Nak(M))\stackrel{\can_M}{\to} k \]

is perfect giving rise to the isomorphism

\[ \Hom_{\cD^b(A)}(M,N)^*\simeq \Hom_{\cD^b(A)}(N,D_\Nak(M)) \]

from <ref>.

Define the
homological duality functor as

\begin{align}\label{Eq:def functor homological duality for alg A}
D_h\colon \cD^+(A\lmod)&\longrightarrow \cD^-(\rmod A)^\op\\
M&\mapsto \RHom_A(M,A)\notag
\end{align}

The same formula sends a right module into a left module.
By abuse of notation we will still write $D_h$ for this functor.

Notice that without further hypotheses, the target category of $D_h$ is only bounded from below.

The functor $D_h$ preserves perfect complexes and is an involution on $\cD^b_\perf(A\lmod\nd)$.

Notice that if $P$ is a finitely generated projective left $A$-module, then $\Hom_A(P,A)$ is a finitely generated projective right $A$-module.
It follows that $D_h$ preserves perfect complexes.

Let us now prove that $D_h^2\simeq \Id$ on perfect complexes.
There is a natural transformation $\ev\colon\Id\to D_h^2$ coming from the evaluation morphism

\[ \ev_P\colon P\to \Hom_A(\Hom_A(P,A),A). \]

Note that $\ev_P$ is compatible with finite direct sums, i.e., $\ev_{\oplus P_i} = \oplus_i \ev_{P_i}$.
Hence in order to show that $\ev_P$ is an isomorphism for every finitely generated projective, it is enough to show it for $P=Ae$, where $e$ is an idempotent.
But then $\Hom_A(P,A)=eA$ and $\Hom_A(eA,A) = Ae=P$ and so clearly $\ev_P$ is an isomorphism.

Since $\ev$ is a natural transformation (functorial) it behaves well on complexes and hence we deduce that $\ev_{P^\bullet}$ is also an isomorphism for every perfect complex $P^\bullet\in\cD^b_\perf(A\lmod\nd)$.

We denote by $\cD^b_{fg}(A\lmod\nd)$, the bounded derived category of finitely generated, non-degenerate, left $A$-modules.
(Similarly for right $A$-modules.)

Suppose $A$ has both left and right finite homological dimension.
Then the homological duality $D_h$ gives a functor

\[ D_h\colon \cD^b_{fg}(A\lmod\nd)\to \cD^b_{fg}(\rmodnd A)^\op \]

whose square is isomorphic to the identity, i.e., $D_h^2\simeq \Id$.

Since $A$ has finite left and right homological dimension, $\cD^b_{fg}(A\lmod\nd) = \cD^b_{\perf}(A\lmod\nd)$, and similarly for right modules.
Now, the conclusion follows from <ref>.

§.§ Grothendieck–Serre duality and homological duality

We provide some general discussion of Grothendieck–Serre and homological duality, and then we apply it in <ref> to the context of smooth representation of $p$-adic groups.

The tools that we will use are the dualizing (or canonical) complex $\omega_A^\circ\in\cD^b(A\lmod)$ for a commutative ring as well as the exceptional pull-back functor $f^!\colon D^+(A\lmod)\to D^+(B\lmod)$ for any finite map of algebras $f\colon A\to B$.
We will provide some of the details without striving for optimal generality.
For a thorough discussion of these matters, one could consult <cit.> or <cit.>.
Check <cit.> for a summary.

For an algebra $R$, we denote by $R\lmod$ the abelian category of left modules over $R$.
We put $\cD^+(R\lmod)$ for the bounded below derived category of $R\lmod$ and $\cD^b(R\lmod)$ for the bounded derived category.
For right modules we use the notation $\rmod R$.
The full subcategory of $R\lmod$ consisting of finite length modules is denoted by $R\lmod^\finl$ and similarly for right modules.

To simplify the discussion of dualizing complexes and upper-shriek functoriality, we start with the following definition:

For the polynomial ring $A=k[X_1,\dots,X_d]$, we put $\omega_A^\circ = A[d]$ as an object in $\cD^b(A\lmod)$ and call it the normalized dualizing complex.

Notice that in general one speaks of dualizing complexes over a ring $A$ (or more generally over a scheme $X$), which, however,  are not unique in general.
In order to make it unique, one normalizes it in such a way to make it compatible with exceptional pullback functors $f^!$, i.e.,  such that $f^!\omega_Y^\circ = \omega_X^\circ$ for any map of  schemes $f\colon X\to Y$.
Since a discussion about $f^!$ for a general map of algebras $f\colon A\to B$ would take us too far afield, we prefer to start by <ref> and only discuss $f^!$ for finite maps.

As a trivial example, notice that for a field $k$, we have $\omega_k^\circ = k[0]$.

If $f\colon A\to B$ is a map of (commutative) algebras, then the natural restriction functor $B\lmod \to A\lmod$ is exact and extends in an obvious way to derived categories.
We will denote this functor by $f_*\colon \cD^b(B\lmod)\to \cD^b(A\lmod)$.
Notice that it can also be described as $f_* = {}_AB_{B}\otimes_B -$ where we have denoted by ${}_AB_B$,  the $A$-$B$-bimodule $B$.

Let $f\colon A\to B$ be a finite map of commutative algebras.
Then the restriction functor $f_*\colon \cD^+(B\lmod)\to \cD^+(A\lmod)$ is left adjoint to
\[ f^!:=\RHom_A(B,-)\colon \cD^+(A\lmod)\to\cD^+(B\lmod). \]

Since $f_* = {}_AB_{B}\otimes -$
the tensor-Hom adjunction gives for a $B$-module $M$ and an $A$-module $N$ the following natural isomorphism:

\[ \RHom_A(B\otimes_B M,N) = \RHom_B(M,\RHom_A(B,N)).\]

In other words, $f^!$ is right adjoint to $f_*$.

The previous proof does not need the map $f$ to be finite.
If the map $f$ is not finite, the right adjoint $\RHom_A(B,-)$ is not denoted by $f^!$.
In general, $f^!$ is right adjoint to a functor $f_!$ (restriction with compact support)  that we will not define (see for example <cit.> or <cit.>).
See <cit.> for a construction of $f^!$ in general.
If $f$ is a finite map, then $f_*=f_!$,  so the right adjoint of $f_*$ deserves the name $f^!$.

Suppose we have defined for a commutative $k$-algebra $A$,  a (normalized) dualizing complex $\omega_A^\circ\in\cD^b(A\lmod)$.
Then for any finite map $f\colon A\to B$ of commutative algebras, we put $\omega_B^\circ:=f^!(\omega_A^\circ)$ and call it the (normalized) dualizing complex of $B$.

(1) This definition of $\omega_B^\circ$ seems to depend on $A$ and on the map $f$.
It turns out that this is not the case, see

(2) By construction, the object $\omega_A^\circ$ has finite injective dimension for the polynomial algebra $A$ because $A$ is smooth.
If $A\to B$ is any finite map, also the dualizing complex $\omega_B^\circ$ will have finite injective dimension.
This can be seen using adjunction and the fact that $B$ has a finite projective resolution as an $A$-module since $A$ is smooth.

(3) Here is a special case of the above considerations. If $R$ is a local Cohen-Macaulay ring with canonical module $\omega_R$, and $A$ is
an algebra over $R$ which is Cohen-Macaulay, and is a finitely generated $R$ module, then if
$c = \dim R - \dim A$,

\[ \omega_A= \Ext^c_R[A, \omega_R],\]

is a canonical module for $A$, and other Ext are zero.

For any ideal $I\le A$, the map $A\to A/I$ is finite, hence $\omega_{A/I}^\circ = \RHom_A(A/I,\omega_A^\circ)$.
In particular, for $\fm$ a maximal ideal of $A$ with residue field $k$, we recover
$\RHom_A(k,\omega_A^\circ) = \omega_k^\circ = k[0]$.

If $A$ is a commutative finite dimensional $k$-algebra, then $\omega_A^\circ = A^*[0]$.
Indeed, using the finite map $k\to A$, and the fact that $\omega_k^\circ = k[0]$,
we deduce immediately the equality
$\omega_A^\circ = \RHom_k(A,k) = A^*[0]$.

As $\omega_A$ has finite injective dimension (see <ref>(2)), the boundedness of the derived categories in the following definition is justified:

For a commutative $k$-algebra $A$ with normalized dualizing complex $\omega_A^\circ$, we define the Grothendieck–Serre duality functor as

\[ D_{GS}:=\RHom_A(-,\omega_A^\circ)\colon \cD^b(A\lmod)\to (\cD^b(A\lmod))^\op.\]

The canonical evaluation morphism $\Id\to D_{GS}^2$ is an isomorphism on finitely generated modules. This follows by the definition of a dualizing complex, see <cit.>.

For $A$ a finite dimensional commutative $k$-algebra we have that $D_{GS} = (-)^*$.
To see this, use  $\omega_A^\circ = A^*[0]$,  discussed in <ref>,  and the tensor-Hom adjunction:

\[ \RHom_A(M,\omega_A^\circ) = \RHom_A(M,\Hom_k[A,k]) = \RHom_k(M,k).\]

For any commutative algebra $A$ with normalized dualizing complex $\omega_A^\circ$,
the functor $D_{GS}$ restricts to a duality

\[ D_{GS}\colon A\lmod^\finl  \to (A\lmod^\finl)^\op\]

which moreover can be identified with the contragredient duality $(-)^* = \Hom_k(-,k)$.

Let $M$ be a finite length $A$-module.
Then there exists a finite codimension ideal $I\le A$ such that the $A$-module structure on $M$ factors through the (finite) map of algebras $A\to A/I$.
Using the right adjoint of restriction (see <ref>) together with the definition of the normalized dualizing complex, we get natural isomorphisms of $A$-modules:

\begin{align*}
D_{GS}(M) & = \RHom_A(M,\omega_A^\circ)\\
& = \RHom_{A/I}(M,\RHom_A(A/I,\omega_A^\circ))\\
& = \RHom_{A/I}(M,\omega_{A/I}^\circ)\\
& = \RHom_{A/I}(M,(A/I)^*)\\
& = M^*
\end{align*}

where in the last two equalities we have used <ref>.

Part of the content of the above lemma is that $D_{GS}(M)$ is concentrated in a single degree if $M$ is a finite length $A$-module.
This is not a priori obvious from the definition of $D_{GS}$ and it is a feature of dualizing complexes.
The fact that it lives in degree $0$ has to do with the normalization that we chose.

In this remark, we recall a part of the  duality theory for Cohen-Macaulay local rings $R$ of dimension $d$ with residue field $k$,
and with a canonical module $\omega_R$ ($\omega_R=R$ if $R$ is Gorenstein)
which we recall has the following two properties:

* $\omega_R$ has finite injective dimension.

* $\dim_k \Ext_R^d(k, \omega_R)= 1.$

Then if $M$ is a Cohen-Macaulay $R$-module of dimension $t$, we have:

* $\Ext^i_R(M,\omega_R)=0$ for $i \not = d-t$.

* $\Ext_R^{d-t}(M, \omega_R)$ is Cohen-Macaulay of dimension $t$.

* $\Ext_R^{d-t}(\Ext_R^{d-t}(M, \omega_R), \omega_R) \cong M$.

We would like to apply all this to non-commutative algebras.
From now on, we consider $R$ a (possibly) non-commutative $k$-algebra together with an algebra map $A\to Z(R)$ from a commutative $k$-algebra $A$ to the center of $R$.
Suppose moreover that $R$ is finite as an $A$-module and that $A$ has a (normalized) dualizing complex $\omega_A^\circ$.
The category of left $R$-modules becomes linear over $A$ and therefore one can consider the following functor

\begin{align*}
D_{GS/A}\colon &\cD^{b}(R\lmod)\to (\cD^{b}(\rmod R))^\op \\
&M\mapsto \RHom_A(M,\omega_A^\circ)
\end{align*}

from the (bounded) derived category of left $R$-modules to the derived category of right $R$-modules.
We will abuse notation and denote by the same symbol the analogous functor from right modules to left modules.

A duality on a $k$-linear category $\cC$ is a $k$-linear functor $D\colon \cC\to \cC^\op$ such that $D^2\simeq \Id$.

Notice that for abelian  and triangulated categories, a duality is necessarily exact.

In the above setting, the functor $D_{GS/A}$ restricts to a duality

\[ D_{GS/A}\colon R\lmod^\finl \to (\rmodfinl R)^\op \]

that can moreover be identified with the contragredient $(-)^*$.

Observe that since $R$ is a finite $A$-module, any finite length $R$-module restricts to a finite length $A$-module.
We can therefore apply <ref> to deduce that $D_{GS/A}(M)\simeq M^*$ as $A$-modules for any $M$ a finite length left (resp., right) $R$-module.
The naturality implies the isomorphism $D_{GS/A}(M)\simeq M^*$ holds also as right (resp., left) $R$-modules.

Notice that the proof above still applies to $R$-modules that are of finite length as $A$-modules
even  if we do not assume that $R$ is finite over $A$.

Recall that for a $k$-algebra $R$, the homological duality was defined in <ref> as the functor

\[ D_h:=\RHom_R(-,R)\colon \cD^\pm(R\lmod)\to \cD^\mp(\rmod R)^\op,\]

where it was shown that it is a duality on perfect complexes.
If moreover $R$ has finite global dimension, then we get a duality on the bounded derived category of finitely generated $R$-modules (see <ref>):

\[ D_h\colon \cD^b_{fg}(R\lmod)\to \cD^b_{fg}(\rmod R)^\op.\]

In general, there is no reason for this functor to preserve any abelian subcategories or to have good properties.
However, in the case of representations of $p$-adic groups it does (see <ref>)  thanks to second-adjointness and the following technical condition that is satisfied on the cuspidal blocks (keeping the assumptions on $R$ and $A$)

(FsG algebra)
Let $R$ be an associative algebra containing a commutative algebra $A$ in its center over which it is finite with $A$ a finitely generated
$k$-algebra for $k$ a field. Then $R$ is called a FsG algebra,
or a  Frobenius symmetric and Gorenstein  algebra, if there is an integer $d$ such that

\[ D_{GS/A}(R)\simeq R[d] \tag{FsG}\]

as  $R$-bimodules.

The   condition (FsG) in Definition <ref>  is satisfied for all cuspidal blocks (see <ref>) but it is not satisfied for all Bernstein
for a  $p$-adic group, for example it does not hold for the Iwahori block for $\SL_2(F)$ as we discuss in
<ref> below.

The name FsG
is inspired by the notions of Frobenius symmetric and Gorenstein algebras.
Recall that a finite dimensional algebra $A$ over a field $k$ is called Frobenius if there exists a nondegenerate
bilinear form $B: A \times A \rightarrow k$ such that $B(ab,c)=B(a,bc)$ for all $a,b,c$ in $A$, and is called Frobenius symmetric if furthermore $B(a,b)=B(b,a)$ for all $a,b$ in $A$. One can see that a finite dimensional $k$ algebra $A$ is Frobenius if and only if there exists a codimension one subspace of $A$ not containing any nonzero right-ideal of $A$. Before proceeding further, recall that if $M$ is a right-$A$ module (resp., left-$A$ module), then
$\Hom_k(M,k)$ is naturally a   left-$A$ module (resp., right-$A$ module).
Now note that
given a bilinear form $B: A \times A \rightarrow k$, the associated map  $A \rightarrow \Hom_k(A,k)$ is equivariant for the right-$A$ module structures if and only if $B(ab,c)=B(a,bc)$ for all $a,b,c$ in $A$, whereas the map
$A \rightarrow \Hom_k(A,k)$ is equivariant for the left-$A$ module structures if and only if $B(ba,c)=B(a,cb)$ for all $a,b,c$ in $A$. Thus a finite dimensional
associative algebra $A$ over $k$ is Frobenius if and only if $\Hom_k(A,k) \cong A$ as right $A$-modules,
whereas $\Hom_k(A,k) \cong A$ is a bi-$A$-module isomorphism if and only if the bilinear form $B: A \times A \rightarrow k$ is both Frobenius and symmetric. If $A$ is commutative and local then  $A$ is Frobenius if and only if $A$ is  Gorenstein. Among non-commutative algebras, the best known Frobenius symmetric algebra is the group algebra
$k[G]$ of a finite group $G$ for $k$ a field of characteristic zero or nonzero.

Keeping the same assumptions, suppose moreover that $R$ satisfies condition (FsG) in Definition <ref>.
Then the (shifted by $d$) homological duality functor $[d]\circ D_h$ is isomorphic to $D_{GS/A}$.
In particular, it restricts to a duality on finite length $R$-modules

\[ [d]\circ D_h\colon R\lmod^\finl \to (\rmodfinl R)^\op\]

that is moreover isomorphic to the contragredient.

Using condition (FsG), tensor-hom adjunction and <ref> we get natural isomorphisms of functors:

\begin{align*}
[d]\circ D_h & = [d]\circ \RHom_R(-,R) \\
& = \RHom_R(-,\RHom_A(R,\omega_A^\circ))\\
& = \RHom_A(-,\omega_A^\circ)\\
& = D_{GS/A}\\
& = (-)^* \qquad\text{ for finite length modules}.\qedhere
\end{align*}

If in Condition (FsG),  we replace $R$-bimodule by left $R$-module, then we still get that $[d]\circ D_h$ sends $R\lmod^\finl$ to $\rmodfinl R$ but we can not identify it with the contragredient.

The above proof gives us a bit more:

Keeping the same assumptions, we have that $[d]\circ D_h \simeq D_{GS/A}$ on all  $R$-modules if and only if condition (FsG) is satisfied.

By the definition of $D_{GS/A}$ and adjunction we have

\begin{align*}
D_{GS/A} &= \RHom_A(-,\omega_A^\circ)\\
& = \RHom_R(-,\RHom_A(R,\omega_A^\circ))
\end{align*}

and by Yoneda's lemma this last functor is isomorphic to $[d]\circ D_h=\RHom(-,R[d])$ if and only if condition (FsG) is satisfied.

Let us consider a particular case that will be of importance to us.
Suppose that $R$ is a finite projective $A$-algebra where $A=k[X_1^{\pm1},\dots,X_d^{\pm1}]$.
We have that $\omega_A^\circ = A[d]$ and then condition (FsG) becomes

\begin{align}\label{Cond: Hom(R,A)=R as bimodules}
\Hom_A(R,A) \simeq R \text{ as $R$-bimodules}.
\end{align}

Keeping the above notation and assuming condition (<ref>), we have the following isomorphisms of functors when restricted to finite-length $R$-modules

\[ \RHom_R(-,R)[d]\simeq \RHom_A(-,A)[d] \simeq \Hom_k(-,k).\]

§ CONVENTIONS ON GROUPS AND REPRESENTATIONS

§.§ Central extensions

Let $G=\bG(F)$
be the locally compact group of $F$-rational
points of a reductive linear algebraic group $\bG$ over a non-archimedean local field $F$.
Let $\wG$ be a finite central extension of $G$ with kernel a finite abelian group ${\mu}$:

\[ 1\to \mu\to\wG\to G\to 1 \]

that is moreover a topological covering.
In this situation it is proved, for example in <cit.>, that $\wG$ admits a
basis of open sets around the identity element of $\wG$ consisting of compact open subgroups lifted from $G$
(i.e., for which the restriction of the map $\wG\to G$ is an isomorphism).
So $\wG$ is a totally disconnected group or an $l$-group in the terminology of [7].

§.§ Representations

In this section we recall some notions around the representation theory of locally compact totally disconnected groups $G$.
All representations will be on complex vector spaces.
One can consult [17, 7, 28] for details.
We will work with the category $\cM(G)$ of smooth complex representations of such a group $G$.

Let $H\le G$ be a closed subgroup of $G$.
Then restricting a representation from $G$ to $H$ we obtain an exact functor between the categories of smooth representations

\begin{align*}
\Res_H^G\colon \cM(G)\to \cM(H).
\end{align*}

The restriction functor has a right adjoint given by induction

\[ \Ind_H^G\colon \cM(H)\to \cM(G)\]

which is defined by,
\[\Ind_H^G(V):=\{f\colon G\to V\mid f(hg) = hf(g),\text{ for all }h\in H \text{ and }g\in G\}^\sm.\]

The pair of functors $\Res_H^G\dashv \Ind_H^G$ has
the adjointness property (for $V$ a representation of $G$, and $W$ of $H$)

\[\Hom_H[\Res_H^G V, W] = \Hom_G[V, \Ind_H^G(W)],\]

which goes under the name of Frobenius reciprocity
for which we refer to  [7] or <cit.>.

The induction functor admits a subfunctor $\ind_H^G\subset \Ind_H^G$ called compact induction consisting of  functions with compact support modulo $H$:

\[ \ind_H^G(V):=\{f\in \Ind_H^G(V)\mid H\backslash\supp(f) \text{ is compact}\}. \]

In case $G/H$ is compact we clearly have $\ind_H^G=\Ind_H^G$.

If $V$ is a smooth representation of $G$ then the contragredient representation of $V$ is defined to be the smooth part of the linear dual $V^\vee:=(V^*)^\sm$.

One can prove (see <cit.>) that induction and compact induction are related to each other through the contragredient.
More precisely, for $V$ a smooth representation of $H$ we have

\[ \Ind_H^G(V^\vee) = \ind_H^G(V\otimes \delta_{H\backslash G})^\vee, \]

where $\delta_{H\backslash G}$ is the modular character of $G$ divided by the one of $H$.

Suppose now that $H\le G$ is open.
Since $H$ is the complement of the union of all non-trivial left cosets $Hg$, $g\in G$, it is also a closed subgroup.
(see <cit.>)
If $H\le G$ is an open subgroup then the restriction functor $\Res_H^G$ is right adjoint to the compact-induction functor $\ind_H^G$.
In particular, $\ind_H^G$ sends projective objects to projective objects.

§.§ Projectives and injectives

We continue with $G$ being a locally compact, totally disconnected group.
Let $\cM(G)$ be the abelian category of smooth representations of $G$.
Let $\Hecke(G)$ be the Hecke algebra of locally constant compactly supported  functions on $G$ endowed with convolution (one needs to choose a left invariant Haar measure on $G$).
The algebra $\Hecke(G)$ is an algebra without unit but with a rich supply of idempotents[It is
what is called an idempotented algebra: for every finite set of element $a_i$, there is an idempotent $e$ such that $ea_i=a_ie=a_i$ for each $a_i$.] because $G$ has a basis of neighborhoods of identity consisting of open compact subgroups.
A representation $V$ of $\Hecke(G)$ is said to be  non-degenerate  if it has the  property that $\Hecke(G)V=V$.
There is the well-known equivalence of the category of smooth representations of $G$ and the category of non-degenerate representations of $\Hecke(G)$:

\[ \cM(G)\simeq \cH(G)\lmod\nd. \]

Suppose that $G$ admits a countable basis of neighborhoods of identity consisting of compact open subgroups.
Then the Hecke algebra $\cH(G)$ is a projective object in $\cM(G)$.
Indeed, one writes $\cH(G) = \bigcup_{i=1}^\infty \cH(G)e_i$ where $e_i=e_{K_i}$ are idempotents corresponding to a countable basis of compact open subgroups of $G$.
Since $ \Hom_G(\cH(G)e_i,V) = V^{K_i}$, each of the $\cH(G)e_i$ is a projective object in $\cM(G)$,
and then one notices that
$\Hom_G(\cH(G), -) = \lim_i \Hom_G(\cH(G)e_i,-)$
and this latter inverse limit is exact because the transition functions are surjective (and hence the Mittag-Leffler condition is automatically satisfied).

The abelian category $\cM(G)$ has a good supply of injective and projective objects, for example
for any open compact  subgroup $K$ of $G$, $\ind_K^G (\bC)$ is a projective object (see <ref>), and its smooth dual $\Ind_K^G (\bC)$ is an injective object.

We use $\Ext^i_{G}(V,V')$ to denote Ext groups in $\cM(G)$.

§.§ Homological dimension

Here $G=\bG(F)$ and $\wG$ is a central extension of $G$ as defined in <ref>.

It is shown in <cit.> that the category of smooth representations of $G$ has finite homological dimension.
The argument uses the building of $G$ to give a resolution
of the trivial module by projective modules which are sums of representations of $G$ of the form  $\ind_K^G (\bC)$ (where $K$ are compact open subgroups of $G$):

\[0\rightarrow P_d \rightarrow  \cdots \rightarrow P_1 \rightarrow P_0 \rightarrow \bC \rightarrow 0,\]

where $d$ is  the split rank of $G$.
Tensoring this resolution with any $G$-module $V$, and observing that
\[\ind_K^G (\bC) \otimes V = \ind_K^G(V|_K),\]

we find a projective resolution of length $\leq d$ for  any $G$-module $V$.  This argument with
the resolution of the trivial module $\bC$ for $G$ works just as well for covering groups
$\wG\to G$ since $\ind_K^G (\bC)$ treated as a $\wG$-module is $\ind_\wK^\wG (\bC)$, and
$\ind_\wK^\wG (\bC) \otimes V = \ind_\wK^\wG(V|_\wK)$ is a projective $\wG$-module.
Therefore,  $\Ext^i_{\wG}(V,V')=0$ for any representations
$V,V'$ of $\wG$,  if $i>d$.

Once we have the Bernstein decomposition for covering groups one can prove exactly as for linear group that the category of smooth representations $\cM(\wG)$ is noetherian (<ref>).
Since it is also of finite homological dimension, any finitely generated module admits a finite resolution by finitely generated projective modules. (If $d$ is the projective dimension of a finitely generated representation $V$, resolve $V$ by finitely generated projective modules  $ P_{d-1} \rightarrow  \cdots \rightarrow P_1 \rightarrow P_0 \rightarrow V \rightarrow 0,$ and then observe as in the proof of Hilbert Syzygy that the kernel of the map $P_{d-1}\rightarrow P_{d-2}$ must be projective.)
This will be useful when we apply the abstract results from <ref> to representation theory in <ref> and <ref> as all finitely generated modules will be perfect.

§ SPLITTING THE CATEGORY OF REPRESENTATIONS

§.§ Compact representations

In this section $G$ denotes an arbitrary  locally compact td-group which is
countable at infinity.
The most important result we need is that compact representations split the category of smooth representations of $G$.

A smooth representation $(\pi,V)$ of $G$ is said to be compact if all its matrix coefficients have compact support.

The existence of a compact irreducible representation
implies that $G$ has compact center.

The first important theorem about compact representations is
that any compact representation which appears as a subquotient in any smooth representation appears as a direct summand by  <cit.>. The following theorem is an easy consequence of it:

Let $\rho$ be a compact irreducible representation of $G$.
Then matrix coefficients of $\rho$ provide us with an injective map of $G\times G$-modules

\[\rho\boxtimes \rho^\vee \subset\cH(G),\]

and  this is the only subquotient of $\cH(G)$ isomorphic to $\rho\boxtimes\rho^\vee$.

Moreover, if $\rho'$ is another irreducible compact representation of $G$, non-isomorphic to $\rho$, then the $G\times G$ representation $\rho\boxtimes\rho'^\vee$ does not appear as a subquotient of $\cH(G)$.

There is a natural map from $\rho\boxtimes \rho^\vee$ to functions on $G$  given by
\[v \otimes \ell \in \rho\boxtimes \rho^\vee  \mapsto F_{v \otimes \ell}(g) = \ell(gv),\]
which if $\rho$ is a compact smooth representation of $G$ lands inside
compactly supported smooth functions on $G$, i.e., in $\cH(G)$. If $\rho$ is furthermore irreducible, the resulting map
$\rho\boxtimes \rho^\vee \subset\cH(G)$ is injective.

As any compact representation which appears as a subquotient in any smooth representation appears as a direct summand by  <cit.>, the rest of the theorem is a direct consequence of
the Frobenius reciprocity:

\[ \Hom_{G \times G}[\ind_{\Delta G}^{ G \times G} (\bC), \rho\boxtimes \rho'^\vee] = \Hom_{\Delta G}[   \rho\boxtimes \rho'^\vee,  \bC] .\qedhere\]

We denote by $\cM(G)_c$ the full subcategory of $\cM(G)$ consisting of those smooth
representations of $G$ all whose irreducible subquotients are compact representations.
It is clearly a subcategory closed under subquotients, direct sums and extensions.

We denote by $\cM(G)_{nc}$ the full subcategory of $\cM(G)$ formed of representations that have no compact irreducible subquotient.

For $\cS$, a collection of irreducible representations of $G$, we denote by $\cM(G)_{[\cS]}$ the subcategory of $\cM(G)$ formed of representations such that all their irreducible subquotients are isomorphic to an object in $\cS$.
Denote by $\cM(G)_{[\text{out }\cS]}$ the subcategory of representations such that none of their irreducible subquotients are isomorphic to an object in $\cS$.

The following theorem summarizes the main properties of compact representations, cf.

For $\cS$ a finite collection of compact irreducible representations of $G$,
we have:

* The category $\cM(G)_{[\cS]}$
is semisimple: all the objects in $\cM(G)_{[\cS]}$ are isomorphic to a direct sum of objects in $\cS$.

* There is  an equivalence of categories

\begin{align}\label{Eq:decompose subquotients in A and outside A}
\cM(G) \simeq \cM(G)_{[\cS]}\times \cM(G)_{[\text {out } \cS]}.
\end{align}

Moreover, the category $\cM(G)_c$ admits a decomposition

\begin{align}
\cM(G)_c = \prod_{\tau} \cM(G)_{[\tau]}
\end{align}

where the product runs over all isomorphism classes of compact irreducible representations of $G$.
In particular, $\cM(G)_c$ is a semisimple category.

It is natural to ask if we can always decompose a representation into a direct sum of compact and non-compact.
This is not completely automatic from <ref>,
and one needs a further finiteness condition:

Consider the following condition (see <cit.>) called "compact finite":

\begin{align}\label{Eq:condition (KF)}
\tag{KF}\quad \begin{minipage}{300pt}
for any compact open subgroup $K\le G$ there is only a finite number of isomorphism classes of compact irreducible representations of $G$ having a non-zero $K$-fixed vector.
\end{minipage}
\end{align}

If the group $G$ satisfies the above condition (KF), then we have a decomposition of categories

\begin{align}\label{Eq:dec compact times non-compact}
\cM(G) = \cM(G)_c\times \cM(G)_{nc}.
\end{align}

Notice that in light of the theorem <ref>, every compact representation is projective-injective in $\cM(G)$.
This is a remarkably strong homological property that will be useful in the sequel.

In our situation, the condition (KF) is satisfied thanks to the uniform admissibility theorem, see <ref>.

§.§ Compact modulo center

We have seen that compact representations behave as nicely as one could hope but for many interesting groups there are no such representations.
It turns out that the issue comes from the non-compactness of the center.
In this subsection we present an analogue of the decomposition of categories <ref> and <ref>.

A representation of $G$ is called compact modulo center
if its matrix coefficients have compact support modulo $Z(G)$.

Denote by $G^\circ$ the subgroup of $G = \bG(F)$,
for $\bG$ a reductive group over $F$, defined by:

\[ G^\circ = \cap\,  \{ {\rm ker}  |\chi|\colon G \rightarrow \bR^+\} ,\]

where $\chi\colon G \rightarrow F^\times$ are the algebraic characters of $G$ defined over $F$.
The subgroup $G^\circ$  of $G$ has the property that $G/G^\circ\simeq \bZ^d$ for some $d\ge 0$.
Further, $G^\circ Z(G)(F)$ is a normal subgroup of finite index in $G(F)$.
The subgroup $G^\circ \subset G(F)$ contains all compact subgroups of $G(F)$ and, in fact, $G^\circ$
is generated by all compact subgroups of $G$. Since we could not find a convenient reference, we provide a proof
for which we thank Sandeep Varma.

The subgroup $G^\circ$ is generated by all compact subgroups of $G$.

Let $G^d=[G,G]$ be the derived subgroup of $G$, and $j: G^s \rightarrow G^d $, the simply connected cover of $G^d$.
The group $G(F)$ operates by conjugation on $G^d$, and this action lifts to $G^s$, thus
$G(F)$ acts on the set of Iwahori subgroups of $G^s(F)$ which are permuted transitively by $G^s(F)$. Let $\I$ be a fixed Iwahori subgroup of $G^s(F)$.
We know that the only elements of $G^s(F)$ that normalize $\I$ belong to  $\I$, and
the stabilizer of $\I$ in $G^\circ$, call it  $N(\I)$, is a compact subgroup of $G(F)$ (this uses the assertion contained in the Lemma for tori which is equivalent to the assertion that anisotropic tori are the same as compact tori). Hence
$N(\I)$ surjects onto $G^\circ/ j(G^s(F))$. It follows that if we can prove the assertion of the Lemma
for the simply connected group $G^s$, then the Lemma follows for the group $G$.

The assertion of the Lemma for the simply connected group $G^s$, assuming that it is isotropic (else $G^s(F)$ is compact
and there is nothing to prove),
follows from the Bruhat decomposition $G^s(F)=\I W^{\rm aff} \I$ where $W^{\rm aff}$ is a Coxeter group which comes equipped
with a set of generators $\{ s \in S\}$ of order 2 where each $s$ treated as an element of $G^s(F)$ is compact
(since $\I \cup \I s \I$ is a compact parahoric subgroup of $G^s(F)$).

We put $\cX(G):=\Hom_\gr(G/\Go, \bC^\times)$ and call it the group of unramified characters of $G$.

A representation $\pi$ of $G$ is compact modulo center if and only if its restriction $\pi|_{G^\circ}$ is compact.

The following well-known proposition, although simple, is the main ingredient allowing one
to pass from $\Go$ to $G$ (since $\Go\cdot Z(G)$ is of finite index in $G$, the proof of this proposition
is the same as the Clifford theory in finite groups):

* Let $(V,\pi)$ be an irreducible representation of $G$.
The irreducible representations of $\Go$ appearing in $\Res_\Go^G(\pi)$ are all conjugate under $G$.
Moreover, the representation $\Res_\Go^G(\pi)$ is semisimple and of finite length.

* Let $(V_1,\pi_1)$ and $(V_2,\pi_2)$ be two irreducible representations of $G$.
Then the  following are equivalent.

* $\Res_\Go^G(\pi_1)\simeq \Res_\Go^G(\pi_2)$,

* there exists an unramified character $\chi\in\cX(G)$ such that

\[ \pi_1\otimes\chi\simeq \pi_2, \]

* $\Hom_{\Go}(\Res_\Go^G(\pi_1),\Res_\Go^G(\pi_2))\neq 0$.

We denote by $\cM(G)_\cs$, the full subcategory of $\cM(G)$ formed of those representations all whose subquotients are compact modulo center.
The set of irreducible objects of $\cM(G)_\cs$ is denoted by $\Irr(G)_\cs$.

We say that two irreducible representations $\rho_1, \rho_2\in\Irr(G)_\cs$ are in the same inertia class if there exists a character $\chi\in\cX(G)$ such that $\rho_1\simeq \chi\rho_2$.
We write the corresponding equivalence relation as  $\rho_1\sim \rho_2$, and denote the inertia class containing $\rho \in \Irr(G)_\cs$ by the square bracket $[\rho]$.
We denote the set of  inertia classes in $\Irr(G)_\cs$ by $[\Irr(G)_\cs]$.

Given $\pi\in\Irr(G)_\cs$, we denote by $\cM(G)_{[\pi]}$, the full subcategory of $\cM(G)$ consisting of those
smooth representations of $G$ whose restriction to $\Go$ have all the irreducible subquotients only among those of $\pi|_\Go$ (a finite set of
compact representations of $\Go$).

<ref> says that two irreducible representations of $G$ are in the same inertia class if and only if their restriction to $\Go$ are isomorphic.
Therefore the objects of the category $\cM(G)_{[\pi]}$ are those smooth representations of $G$
all of whose irreducible subquotients are in the same inertia class as $\pi$.

Using the above proposition and <ref>, one immediately deduces:

For an irreducible compact modulo center representation $\pi$ of $G$,
we have a decomposition of the category $\cM(G)$ as

\[ \cM(G)\simeq \cM(G)_{[\pi]}\times \cM(G)_{[\text{out }\pi]}. \]

Moreover, the category $\cM(G)_\cs$ decomposes as

\[\cM(G)_\cs =\prod_{[\pi]\in\Irr(G)_\cs} \cM(G)_{[\pi]}.\]

The full subcategory of $\cM(G)$ formed of those representations that have no irreducible subquotient that is compact modulo center is denoted by $\cM(G)_\indu$.
Using <ref> and <ref> one deduces easily:

If the group $\Go$ satisfies condition (<ref>) then we have a decomposition of categories

\[ \cM(G) = \cM(G)_\cs\times \cM(G)_\indu.\]

§.§ The simplest Bernstein component

The goal of this subsection is to formulate and prove an analogue of <ref>

Fix $\pi\in\cM(G)$ an irreducible, compact modulo center representation of $G$.
Then $\pi\boxtimes \pi^\vee\in\cM(G\times G)$ is irreducible and compact modulo center.
We consider the Hecke algebra $\cH(G) = \ind_{\Delta G}^{G\times G}\bC$ as a $G\times G$-module and we try to understand the part of $\cH(G)$, denoted by $\cH(G)[\pi\boxtimes\pi^\vee]$, that lives in $\cM(G\times G)_{[\pi\boxtimes\pi^\vee]}$ (see <ref>).

Let us fix $\pi_0\subset \pi\mid_\Go$, an irreducible representation of $\Go$.
Put $G_1 = \{g\in G\mid \pi_0\simeq {}^g\pi_0\}\le G$.
It is a normal subgroup of $G$  containing  $\Go Z(G)$
which is of  finite index in $G$, hence $G_1$ is of finite index in $G$.
Denote by $\Sigma$ the group $G/G_1$ and its order by $f$.

Put $H:=\Delta(G) (\dGo)\le G\times G$.
The following is the main result of this subsection:

For $\pi\in\cM(G)$ an irreducible, compact modulo center representation of $G$,
we have an isomorphism of $G\times G$ representations

\begin{align}\label{Equation 1}
\cH(G)[\dpi]\simeq \ind_{\Delta(G_1)(\dGo)}^{G\times G}(\pi_0\boxtimes \pi_0^\vee).
\end{align}

\begin{align}\label{Equation 2}
\begin{array}{rl}(\dpi)\otimes \ind_H^{G\times G}\bC &\simeq \Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go}) \otimes  \cH(G)[\dpi]\\
&\simeq e^2f \cH(G)[\dpi],
\end{array}
\end{align}

where $ \Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go})$ is a finite dimensional representation of $G/\Go$, treated as a representation
of $G\times G$ trivial on $H=\Delta(G) (\dGo)\le G\times G$.

Put $\cS = \{\pi' \mid \pi'\subset \pi\boxtimes\pi^\vee |_{G^\circ\times G^\circ} \text{ irreducible}\}$.
The set $\cS$  is a finite set of compact irreducible representations of $\dGo$ and given <ref> we can write

\[ \cH(G)|_{\dGo} = \cH(G)_{[\cS]}\oplus \cH(G)_{[out\,\cS]}\]

as $\dGo$-representations.
Moreover, since $\cS$ is stable under conjugation by $G\times G$, the above decomposition holds as $G\times G$-modules.
Let us write $\cH(G){[\pi\boxtimes \pi^\vee]}$ for the $G\times G$ representation $\cH(G)_{[\cS]}$.

For $\sigma\in\Sigma$, let $\pi_\sigma:=\sigma\pi_0\subset \pi$, an irreducible
sub $\Go$-representation of $\pi|_\Go$.
Notice that $\pi_\sigma$ is isomorphic to the conjugated representation ${}^\sigma\pi_0$ where the action on $\pi_0$
is conjugated by $\sigma$.
The irreducible summands of $\pi|_\Go$ are isomorphic to $\pi_\sigma$ for some $\sigma\in\Sigma$ and $\pi_\sigma\not\simeq\pi_{\sigma'}$ if $\sigma\neq\sigma'$.
Let's write the restriction of $\pi$ to $\Go$ as (for an integer $e\geq 1$):

\[ \pi|_\Go = \bigoplus_{\sigma\in\Sigma} e \pi_\sigma. \]

For an irreducible representation $\pi$ of $G$, the set of unramified characters $\chi: G/\Go\to \bC^\times$ with $\pi \otimes \chi \simeq \pi$ is a finite abelian group, call it $X(\pi)$.  Then, the space $\Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go}) $ comes equipped with an action of the abelian group $G/\Go$, diagonalizing which
gives a canonical basis (up to scalars) which are nothing but the intertwining operators $\pi \otimes \chi \simeq \pi$, thus one sees that  $\dim \Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go}) = e^2f$
is the number of  the intertwining operators $\pi \otimes \chi \simeq \pi$, and
$\Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go})$ comes equipped with a basis $e_\chi$ such that $g\cdot e_\chi = \chi(g)e_\chi$.

The $\Go$-representation $\pi_0$ does not extend to a representation of $G_1$ but the $(\dGo)$-representation $\dpio$ does extend to a representation of $\Delta(G_1) (\dGo)$: this is most easily seen by noticing that $\pi_0$ extends to a projective
representation of $G_1$, and therefore $\pi_0\boxtimes \pi_0^\vee$ is canonically a representation of $\Delta(G_1)$,
hence of  $\Delta(G_1) (\dGo)$.
The representations $\pi_\sigma\boxtimes \pi_{\sigma'}^\vee$ are also representations of $ \Delta(G_1) (\dGo) $ as
they can be written as $(\sigma,\sigma')\dpio$.

In what follows, let $\tau$ be the irreducible representation of $H=  \Delta(G)(\dGo),$

\[\tau =   \bigoplus_{\sigma \in \Sigma} \pi_\sigma \boxtimes \pi_{\sigma}^\vee = \ind_{ \Delta(G_1)(\dGo)}^{H}  (\pi_0\boxtimes \pi_{0}^\vee) .\]

By induction in stages: $G \times G \supset {\Delta(G)}(\Go \times \Go) \supset {\Delta(G)}$,
we can write:

\[ \cH(G) = \ind_{\Delta(G)}^{G\times G} \bC = \ind_H^{G\times G}\cH(\Go), \]

as a representation of $G\times G$, where we recall that $H=\Delta(G) (\dGo)\le G\times G$
is a normal subgroup of $G\times G$ with a natural action on $\cH(\Go)$
(where $\Delta(G)$ acts on $\Go$ by the conjugation action, and $\dGo$ acts on $\Go$ by left and right translations).
Since the set $\cS$ is stable under conjugation by $G\times G$, we deduce the following

\[ \cH(G)[\dpi] = \ind_H^{G\times G}(\cH(\Go)_{[\cS]}).\]

<ref> tells us that in $\cM(\dGo)$, we have a natural isomorphism

\[\cH(\Go)_{[\cS]} \simeq \bigoplus_{\sigma\in\Sigma} \dpis \simeq
\ind_{\Delta(G_1)(\dGo)}^{ \Delta(G)(\dGo) }(  \pi_0\boxtimes \pi_{0}^\vee        ),\]

which is moreover an isomorphism of $H$-modules, proving the isomorphism (<ref>).

All the irreducible subrepresentations of $\dpi|_H$ are of the form $(\sigma,1)\tau\simeq {}^{(\sigma,1)}\tau$, hence

\begin{align}\label{Eq:dpi restricted to H}
\dpi|_H = \bigoplus_{\sigma\in\Sigma}e^2 (\sigma,1)\tau.
\end{align}

Put these together and use the Mackey's relation $V\otimes \ind_K^{G} W = \ind_K^G(V|_{K}\otimes W)$ to get

\begin{align*}
(\dpi)\otimes \ind_H^{G\times G}(\bC)  & \simeq \ind_H^{G\times G}(\dpi|_H)\\  & \simeq
e^2f\ind_H^{G\times G}(\tau)  \\ & \simeq e^2f \ind_{ \Delta(G_1)(\dGo)}^{ G \times G }  (\pi_0\boxtimes \pi_{0}^\vee) \\
& \simeq
\Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go})
\otimes  \ind_{ \Delta(G_1)(\dGo)}^{ G \times G }  (\pi_0\boxtimes \pi_{0}^\vee) \\
& \simeq
\Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go}) \otimes \cH(G)[\dpi],
\end{align*}

where in the 2nd isomorphism above, we have used (<ref>)
together with  the fact that $H$ is  a normal subgroup of $G \times G$, and  $\ind_H^{G\times G}(\tau) =  \ind_H^{G\times G}(\tau^{(\sigma, 1)})$
for all $\sigma \in \Sigma$; the last isomorphism is the  isomorphism (<ref>), and the isomorphism
previous to that follows
since $\Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go})$ consists of characters of $G/\Go$, treated as a
character of $G\times G$ trivial on $\Delta(G)(\Go \times \Go)$. Thus we have  proved the isomorphism (<ref>).

The isomorphism (<ref>) of
the above proposition  can be interpreted as saying that for the cuspidal representation $\pi \boxtimes \pi^\vee$ of $G\times G$,
the corresponding Bernstein component in $\cH(G)$ “contains” all representations
$(\pi\alpha)  \boxtimes (\pi \alpha)^\vee$ where $\alpha$ runs over all the unramified characters of $G/\Go$.
Since  $\pi \otimes \chi \simeq \pi$ for $e^2f$ many characters, this number shows up as multiplicity in the right hand side of (<ref>).
This means that  for each irreducible cuspidal representation $\pi$ of $G$,
$\pi \boxtimes \pi^\vee$ “appears” in $\cH(G)$ with multiplicity 1, where the precise meaning of  “appears” is to be understood in the derived sense, see <ref>.

Suppose the group $G$ is either a reductive $p$-adic group, or a finite cover of it.
Then for a standard parabolic $P=LN$ inside $G$, and any cuspidal pair $[L, \rho]$
defining a Bernstein block $\fs$ (see <ref>), similar to <ref>, it is natural to propose the following isomorphism in $\cM(G\times G)$

\[\ind^{ G \times G}_{P \times P^-}(\rho \boxtimes \rho^\vee \otimes \ind_{\Delta(L) (L^o \times L^o)}^{ L\times L} (\bC)) \simeq e^2fw \cH(G)[\fs \times \fs^\vee],\]

where $e,f$ are defined as before
for the cuspidal representation $\rho$ of $L$, and $w$ is the order of $N_G(L, [\rho])/L$ where  $N_G(L, [\rho])$ is the normalizer
of $L$ preserving $\rho$ up to an unramified character. This will be a kind of Plancherel decomposition in the smooth category,
see [21] for some results in this direction.

The following corollary is a consequence of the proof of the above proposition.

The component of $\cH(G)$ in $\cM(G)_{[\pi_1\boxtimes\pi_2^\vee]}$ is zero unless $\pi_2\simeq\pi_1\chi$ for some unramified character $\chi$ of $G$.

In the language of Bernstein decomposition reviewed in <ref>, this is equivalent to saying that the Bernstein component of $\cH(G)$ corresponding to $\pi_1\boxtimes\pi_2^\vee$ is zero unless $\pi_2\simeq\pi_1\chi$ for some unramified character $\chi$ of $G$.

We note the following corollary that will be useful to us in <ref> when studying contragredients.
It also appears in <cit.>

For any irreducible representation $\pi\in\cM(G)$, compact modulo center, we have an isomorphism of $G$-modules

\[ \Hom_G(\ind_{\Go}^G(\pi|_{\Go}),\cH(G))\simeq \ind_{\Go}^G(\pi^\vee|_{\Go}),\]

where in the left hand side, one considers $\Hom_G(-, \cH(G))$
for the left $G$-module
structure on $\cH(G)$, and as $\cH(G)$ is both left and right $G$-module,
$\Hom_G(-, \cH(G))$ continues to be a right $G$-module.

By the Mackey theory, since $(\Go \times G) H = G \times G$, it follows that
the restriction of $\ind_H^{G\times G}(\bC)$ to    $\Go \times G$ is $\bC \otimes   \ind_{\Go}^G\bC$.
Hence by
the isomorphism (<ref>), and Frobenius reciprocity:

\begin{eqnarray*}
e^2f\Hom_G(\ind_{\Go}^G(\pi|_{\Go}),\cH(G)) & \simeq &
e^2f\Hom_G(\ind_{\Go}^G(\pi|_{\Go}), \cH(G)[\dpi]) \\
& \simeq & \Hom_G(    \ind_{\Go}^G(\pi|_{\Go}),   \pi \otimes \pi^\vee \otimes  \ind_H^{G\times G}(\bC)    ) \\
& \simeq &
\Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go} \otimes (\pi^\vee \otimes  \ind_{\Go}^{G}\bC) \\
& \simeq & \Hom_{\Go}(\pi|_{\Go},  \pi|_{\Go}) \otimes (\pi^\vee \otimes  \ind_{\Go}^{G}\bC) \\
&\simeq & e^2f \ind_{\Go}^G(\pi^\vee|_{\Go}),
\end{eqnarray*}

where in the 4th isomorphism we have used the following lemma.

Let $H_1$ (resp., $H_2$) be a totally disconnected group, and $\pi_1$ (resp., $\pi_2$) any
smooth representation of $H_1$ (resp., $H_2$).
If $\pi_1$ has finite length, then treating the representation $\pi_1 \otimes \pi_2$ of $H_1 \times H_2$
as a representation of $H_1 \subset H_1 \times H_2$, we have,

\[\Hom_{H_1}(\pi_1,\pi_1 \otimes \pi_2) \simeq \Hom_{H_1}(\pi_1,\pi_1) \otimes \pi_2\]

as representations of $H_2$.
The following lemma will be used in the next proposition.
Let $G$ be a totally disconnected group, $G^o$ a normal subgroup of $G$ such that $G/G^o$ is a discrete group. Let $M, N$ be two smooth representations of $G$. Assume that $M$ is an irreducible
representation of $G$ which restricted to $G^o$ is a finite sum of irreducible representations
which are                both injective and projective as a smooth representation of $G^o$. Then,

\[  \Ext^i_{G}[M,N] =  H^i(G/\Go, \Hom_{\Go}[M,N]).\]

\[\hspace {5cm}\cdots  \rightarrow P_1  \rightarrow P_0  \rightarrow  M  \rightarrow 0, \hspace {2.9cm} (1)\]

be a projective resolution of $M$ as a $G$-module. Since $M$ restricted to $G^o$
is both an injective and a projective representation of $G^o$, assume that $P_i$ restricted to $G^0$ is a direct sum of
copies of the irreducible components of $M$ restricted to $G^o$.

The exact sequence (1) gives rise to an exact sequence of $G/G^o$-modules (the exactness follows since we are assuming as discussed in the last paragraph that $P_i$ are projective modules restricted to $G^o$):

\[ 0\rightarrow \Hom_{G^o} [M, N] \rightarrow \Hom_{G^o} [P_0, N]  \rightarrow \Hom_{G^o} [P_1, N]  \rightarrow  \Hom_{G^o}[ P_2, N]
\rightarrow \cdots \hspace{.1cm}(2). \]

To prove the lemma is suffices to prove that for $P$ a projective module for $G^0$, and $N$ arbitrary smooth module for
$G$, $\Hom_{G^0}[P,N]$ is an injective module for $G/G_o$ (as then the exact sequence (2) will be an injective resolution of $\Hom_{G^o} [M, N]$ as $G/G^o$-modules).  Injectivity of $\Hom_{G^o} [P, N]$ as a $G/G^o$-module
is a simple consequence of the adjunction (for $Q$ any module for $G/G^o$):

\[ \Hom_G[Q, \Hom_{G^o}(P, N)] = \Hom_G[Q \otimes P, N] = \Hom_G [P, \Hom(Q, N)]. \qedhere \]

The next result, based on <ref>, seems worth including as its proof is fairly elementary.
A generalization of this proposition to finite length representations plus functoriality is proved in <ref>.
Recall that $G/\Go\simeq \bZ^d$ for some integer $d\ge 0$.

Let $\pi$ be an irreducible, compact modulo center representation of $G$.
Then $\RHom_{G}^{\bullet} (\pi, \cH(G)),$ lives only in degree $d\geq 0$ (where $d$ is the integer for which $G/\Go\simeq \bZ^d$), where it is isomorphic to $\pi^\vee$.

By <ref>,
the proof of this proposition boils down to proving that (for $H= \Delta(G)(G^o\times G^o)$),
$\RHom_{G}^{\bullet} (\pi, \pi \boxtimes \pi^\vee \otimes \ind_H^{G\times G}(\bC))$
lives only in degree $d$ and

\[\Ext_{G}^{d} (\pi, \pi \boxtimes \pi^\vee \otimes \ind_H^{G\times G}(\bC)) = (e^2f)\pi^\vee.\]

We apply Lemma <ref>  to $M=\pi$ (treated as a module for $G\times G$
on which $1 \times G$ acts trivially), and $N= \pi \boxtimes \pi^\vee \otimes \ind_H^{G\times G}(\bC)$.
Note that $ \ind_H^{G\times G}(\bC)$ restricted to $\Go \times G $ is $\ind_{\Go \times \Go}^{\Go \times G }(\bC)$.		Thus, $N$ restricted to $\Go \times G \subset G \times G$
is $\pi|_{\Go} \otimes \pi^\vee \otimes \ind_{\Go}^{G}(\bC)$.

\[ \Hom_{\Go}[\pi , \pi \otimes \pi^\vee \otimes \ind_H^{G\times G}(\bC) ] =\Hom_{\Go}[\pi , \pi] \otimes \pi^\vee
\otimes \ind_{\Go}^{G}(\bC) ,\]

as a module for $(G/\Go) \times G$ where the actions of $G/\Go$ and $1 \times G$
coincide on $\ind_{\Go}^{G}(\bC)$.
\[H^i(G/\Go, \Hom_{\Go}[M,N]) =H^i(G/\Go, \Hom_{\Go}[\pi , \pi] \otimes \ind_{\Go}^{G}(\bC)) \otimes \pi^\vee.\]

Now we know that  $\Hom_{\Go}[\pi , \pi]$ is a representation of $G/\Go$ of dimension $e^2f$, therefore,

\[\Hom_{\Go}[\pi , \pi] \otimes \ind_{\Go}^{G}(\bC) \simeq e^2f  \ind_{\Go}^{G}(\bC) ,\]

both as a module for $G\times 1$ and $1 \times G$.

Hence we are reduced to understanding $H^i(\bZ^d, \bC[\bZ^d])$ which is well-known to be zero for $i<d$, and
$H^d(\bZ^d, \bC[\bZ^d]) = \Ext_{\bZ^d}^d(\bC, \bC[\bZ^d]) \simeq \bC$, completing the proof of the Proposition.

Later, in Corollary <ref>, we will prove that Proposition <ref>
remains true for any finite length cuspidal representation of a $p$-adic group or of a finite cover of it. We take this occasion to mention that the Bernstein block even of a cuspidal representation is not totally trivial:
although it is true that  a finite length indecomposable cuspidal representation of $\wG$ is the successive
extension of a fixed irreducible cuspidal representation $\rho$ of $\wG$,
it is not true
that such a finite length indecomposable cuspidal representation of $\wG$ is of the form $\rho \otimes \lambda$
$\rho$ is an irreducible cuspidal representation of $\wG$,
$\lambda$ is a finite dimensional indecomposable representation of $\wG/\wGo$ (which is the case when $\rho|_{\wGo}$ is irreducible, in which case, indeed this Bernstein block is rather simple).

§ BASIC NOTIONS OF REPRESENTATION THEORY

We place ourselves in the setting $G=\bG(F)$ for $\bG$ a reductive group over a non-archimedean local field $F$ and $\wG$ a covering group of $G$ (see <ref>).
All the basic notions for linear groups have an analogue for the covering group $\wG\to G$.

§.§ Parabolics, Levi and the Weyl group

A parabolic subgroup $\wP$ for $\wG$ is simply the preimage of a parabolic subgroup $P$ of $G$.
Similarly for Levi subgroups and tori[Notice that $\wT$ is not necessarily abelian but this is immaterial to us.] in $\wG$.
A Levi decomposition $P=LN$ for $P\le G$ lifts to a Levi decomposition $\wP = \wL N$ where $N\le \wG$ is the unique lift of $N$ to $\wG$ that is normalized by $\wL$ (in characteristic zero such a lift is obvious, in general see <cit.>).

For $P$ a parabolic subgroup of $G$ with Levi decomposition $LN$ we denote by $P^-$ the opposite parabolic with Levi decomposition $LN^-$.
Similarly for their preimages in $\wG$.

The Weyl group of $\wG$ is defined simply to be equal to $W$, the Weyl group of $G$.

§.§ Parabolic induction and restriction

The notion of parabolic restriction (Jacquet modules), and parabolic induction, make sense for  $\widetilde{G}$, and they have the same adjointness properties as in the linear case.

Let $\wP = \wL N$ be a parabolic with a Levi decomposition.
The (normalized) functors of parabolic induction and parabolic restriction are defined as follows (for details see <cit.> whose convention for the modulus character is opposite that of [17] and [18], which explains
interchange of the factors $\delta_\wP^{\pm 1/2}$ of
[17] and [18]
with their inverses here):

\begin{align*}
\bfi_{\wL,\wP}^\wG\colon &\cM(\wL)\to \cM(\wG) \\
&\pi\mapsto \ind_{\wP}^{\wG}(\pi\otimes\delta_\wP^{-1/2})\\
\bfr_{\wL,\wP}^{\wG}\colon& \cM(\wG)\to \cM(\wL) \\
&\tau\mapsto (\Res_{\wP}^\wG (\tau)\otimes\delta_{\wP}^{1/2})_N
\end{align*}

where $(-)_N$ is the functor of coinvariants under $N$ and $\delta_\wP\colon \wP\to \bR^\times_{+}$ is the modular character of $\wP$.
Both functors are exact.

Frobenius reciprocity states that $\bfi_{\wL,\wP}^{\wG}$ is right adjoint to $\bfr_{\wL,\wP}^\wG$.
As a consequence, parabolic induction preserves limits and parabolic restriction preserves colimits.

It is easy to establish that parabolic induction preserves admissibility (see <cit.>): this is because the double-coset  $\wK\backslash \wG/\wP$ is finite for any open compact $\wK\le \wG$.
The analogous result for parabolic restriction is not so immediate (see <ref>).

It is also rather straightforward to show that parabolic restriction preserves finite type representations.
However, the analogous result for parabolic induction is not so trivial and the proof is based on Bernstein's decomposition <ref>.

We define cuspidal representations of $\widetilde{G}$ just as in the linear case: those smooth representations of
$\widetilde{G}$  whose all nontrivial parabolic restrictions vanish (i.e., all nontrivial Jacquet modules vanish).

By a similar argument as in the linear case (see <cit.>), the geometric lemma holds for
covering groups $\wG$ too
and allows one to calculate the Jordan–Hölder series of the Jacquet modules of a parabolically induced representation of $\widetilde{G}$.
Then one can conclude as in the linear case that every irreducible representation of $\widetilde{G}$
is a subquotient of a parabolically induced  representation of $\widetilde{G}$ induced from a cuspidal representation of a Levi subgroup of $\widetilde{G}$.

§.§ Iwahori decomposition

A compact open subgroup $K$ of $G$ (or of $\wG$) is said to have an Iwahori factorization with respect to a parabolic
$P=LN$ with opposite parabolic $P^-=LN^-$ if the natural map given by multiplication:

\[ m\colon K^+ \times K^0 \times K^- \rightarrow K,\]

is a bijection, where

\begin{align*} K^+ &=  K \cap N, \\
K^0 &=  K \cap L ,\\
K^- &=  K \cap N^-.
\end{align*}

For central extensions $\wG$, we replace $P=LN$ with $\wP=\wL N$ and $\wP^-=\wL N^-$ and everything makes sense to define Iwahori factorization in $\wG$.

§.§ Cartan decomposition

The Cartan decomposition for the group $G$ plays an important role in the proof of the uniform admissibility theorem.
The analogous theorem for $\wG$ is proved by taking the inverse image from $G$.
Let us recall some notions before we state the theorem.

Recall the subgroup $\Go\le G$ from <ref>.
$\wGo$ be the subgroup of $\wG$ which  is the preimage of $\Go\le G$.
We put $\Lambda(\wG):=\wG/\wGo = G/\Go$; it is a finite rank free $\bZ$-module.

The set of unramified characters of $\wG$ is defined to be

\[ \cX(\wG):=\Hom(\wG/\wGo,\bC^\times) = \Hom(G/\Go,\bC^\times). \]

It is clear  that the image of $Z(\wG)$ has finite index in $\wG/\wGo$ and hence the rank of
$Z(\wG)$ is the same as the rank of $Z(G)$.
In other words, we have $\cX(\wG)\simeq (\bC^\times)^{d_G}$ where $d_G$ is the split rank of $Z(G)$.
Notice that by definition, the unramified characters of $\wG$ are the same as those of $G$ (identified through the map $\wG \to G$).

For $\bL$, a reductive algebraic group over $F$, we denote by $A_\bL$, the maximal split torus contained in the center of $\bL$, and denote
$d(\bL) :=\dim(A_\bL)$, and call it the split rank of the center of $\bL$.
We extend this definition to any finite central extension $\wL$ of $L=\bL(F)$, defining
$d(\wL)=  d(\bL) =\dim(A_\bL)$.

Put $A:=A_\bL(F)$ and notice that the image of $A/A^\circ \to L/L^\circ$ has finite index.
Moreover, the surjection $A\to A/A^\circ$ admits a section that is a group homomorphism (use a uniformizer in $F$) and so we get an injection $A/A^\circ \hookrightarrow L$ whose image in $L/L^\circ$ has finite index.

If $\wL$ is a finite central extension of $L$, then the natural map $Z(\wL)\rightarrow Z(L)$ induces a map of lattices $\Lambda(Z(\wL))\to \Lambda(Z(L))$ whose image has finite index.
We can therefore find a lift $\Lambda(Z(\wL))\to \wL$ that is a group homomorphism and moreover its image in $\Lambda(\wL)=\Lambda(L)$ has finite index.
Summarizing we have a diagram

\[ \xymatrix{
\wL \ar[r]& \Lambda(\wL) \ar[r]^{=}  & \Lambda(L)\\
Z(\wL)\ar@{^(->}[u] \ar[r]& \Lambda(Z(\wL)) \ar@{^(->}[r] \ar@{^(->}[u]& \Lambda(Z(L)) \ar@{^(->}[u]
} \]

and from the discussion above we have a section $\Lambda(Z(\wL))\to Z(\wL)$ which is a group homomorphism.
Denote its image by $C_A$; it is a central subgroup of  $\wL$.

Fix also finitely many elements $F_A:=\{f_1,\dots,f_r\in \wL\}$
that lift some system of representatives for $\Lambda(\wL)/\Lambda(Z(\wL))$.

All in all, we have that $C_A F_A$ provides a set-theoretic lift of $\Lambda(\wL)$ to $\wL$ that is moreover a group homomorphism when restricted to $\Lambda(Z(\wL))$.

Apply the above discussion to a parabolic subgroup $\wP=\wL N$ and recall the notion of dominant cocharacters of $L$ with respect to $P$.
Let $L^+$ denote the preimage of dominant cocharacters of $L$ through the map $L\to \Hom_\gr(L,\bG_m)^*$ given by $m\mapsto (\chi\mapsto \chi(m))$.
We let $\wL^+$ denote its preimage in $\wL$ under $\wL\to L$
We can choose the above $F_A$ to lie in $\wL^+$ and similarly we can consider $C_A^+\subset C_A$ the subset of dominant elements in $C_A$.

Let $P_0$ be a fixed minimal parabolic subgroup of $G$ with Levi decomposition $P_0=L_0N_0$ and denote by $\wP_0 = \wL_0 N_0$ the corresponding subgroups in $\wG$.

Let $A=A_0$ be the split component of $P_0$ and $C_AF_A\subset \wL_0$ as above.
(Cartan decomposition)
There exists a maximal compact subgroup
$K_0$ of $G$ such that for its inverse image
$\wK_0$ inside $\wG$, we have:

* $\wG = \wP_0 \wK_0 = \wK_0\wP_0$

* $\wG = \bigsqcup_{af\in C_A^+F_A} \wK_0 af\wK_0$.

Put $\cH_0 = \cH(\wK_0,\wK)$ to be the Hecke algebra of $\wK$-biinvariant functions on $\wK_0$
where  $\wK\le \wK_0 $ is any compact open subgroup of $\wK_0$.
It is a finite dimensional algebra over $\bC$.

A rather easy consequence of this theorem, which is essential in the proof of the uniform admissibility
theorem (<ref> below), is the following decomposition of the Hecke algebra:

With the notation as above, let $K\le K_0 \le G$ be  compact open subgroups of $G$, and
$\wK\le \wK_0 \le \wG$ be the compact open subgroup of $\wG$ obtained by taking their
inverse images in $\wG$.
Then the Hecke algebra $\cH(\wG,\wK)$ decomposes as

\[ \cH(\wG,\wK) = \cH_0 D\cC\cH_0 \]

where $D$ is a vector subspace spanned by functions indexed by $F_A$ and $\cC$ is a subalgebra isomorphic to the group algebra of $C_A\simeq \Lambda(Z(\wL_0))$.

§ BASIC THEOREMS

As before, $\wG$ is a covering group (see <ref>) of a reductive $p$-adic group $G$, where $G=\bG(F)$ for $\bG$ a reductive group over $F$, a non-archimedean local field.
All the basic theorems concerning representations of reductive $p$-adic groups hold also for $\wG$ with the same proofs.
We will give precise references to [28] where the analogous results for $G$ are proved.

The groups $G$ and $\wG$ do not have any compact representations in general because their centers may be non-compact.
However, we can ask for the next best thing (see <ref>):

An irreducible smooth  representation $(V,\pi)\in \cM(\wG)$ is called compact modulo center if all its matrix coefficients have compact support in $\wG/Z(\wG)$.

The classical theorem of Harish-Chandra still holds with the same proof:

An irreducible smooth
representation $(\pi,V)$ of $\wG$ is cuspidal if and only if it is compact modulo center.

Using the easy fact that parabolic induction preserves admissibility, the fact that all compact representations are admissible and the above theorem we can prove the admissibility of irreducible representations (for a proof in the linear case see for example <cit.> or <cit.>):

Any irreducible representation of $\wG$ is admissible.

Not only irreducible representations are admissible but in fact they are so in a uniform way.
Below is the precise formulation of the Uniform Admissibility Theorem due to Bernstein whose proof is based on the decomposition of the bi-invariant Hecke algebra given in <ref> and some tricky linear algebra:

Let $\wK\le \wG$ be an open compact subgroup.
There exists a constant $c=c(\wK)$ such that for all irreducible representations $(\pi,V)$ of $\wG$ we have
$\dim(V^\wK)\le c$.

One can also look at <cit.> for a proof.

For establishing the Bernstein decomposition, and prior to that, verifying the condition (<ref>) which leads to <ref>, one makes use of the following

Given an open compact subgroup $\wK\le \wG$ there is a compact modulo center subset $\Omega\subset \wG$ such that for any irreducible cuspidal representation $(V,\pi)$ of $\wG$ and any vector $v\in V^\wK$ the function

\[ 	g\mapsto  e_{\wK}\cdot\pi(g)(v)\]

has support contained in $\Omega$.

The following theorem, which goes under the name of  <<the geometric lemma>>,
is extensively used.
It is due to Bernstein and Zelevinsky <cit.>.

Let $\wP=\wM N$ and $Q=\wL U$ be two parabolic subgroups of $\wG$ with Levi decompositions.
Let $\rho$ be an irreducible cuspidal representation of $\wM$ and put $\tau:=\bfr_{\wL,\wQ}^\wG \bfi_{\wM,\wP}^\wG(\rho)$.
Then we have

* If $\wL$ has no Levi subgroup conjugate to $\wM$, then $\tau=0$.

* If $\wM$ is not conjugate to $\wL$, then $\tau$ has no cuspidal subquotient.

* If $\wM$ and $\wL$ are standard and conjugate,
then $\tau$ has a filtration with subquotients isomorphic to ${}^w\rho$ for $w\in W(\wL,\wM)/W_\wL$.
In particular, $\tau$ is cuspidal.

In the above statement $W_\wL$ is the Weyl group of $\wL$ and $W(\wL,\wM)$ is the subset of the Weyl group of $\wG$ conjugating $\wL$ into $\wM$.

The next result, proved using the above theorem, shows that the Jordan–Hölder factors of a parabolically induced representation of an irreducible cuspidal are independent of the chosen parabolic and depend only on the conjugacy
class of the cuspidal datum.

Let $\wP=\wL N$ and  $\wP'=\wL' N'$ be two parabolic subgroups in $\wG$ with Levi decompositions.
Let $\rho\in\Irr_\cs(\wL)$ and $\rho'\in\Irr_\cs(\wL')_\cs$ and $\pi = \bfi_{\wL,\wP}^\wG(\rho)$ and $\pi' = \bfi_{\wL',\wP'}^\wG(\rho')$ be the induced representations.
Then $\pi$ has finite length and the following are equivalent.

* The cuspidal data $(\wL,\rho)$ and $(\wL',\rho')$ are conjugate,

* The Jordan–Hölder series of $\pi$ and $\pi'$ are equivalent,

* The Jordan–Hölder series of $\pi$ and $\pi'$ have a common element.

In particular, if $\wL=\wL'$ and $\rho=\rho'$, this shows that $\pi$ and $\pi'$ have the same Jordan–Hölder series, independent of the chosen parabolic.

We can now deduce

Parabolic induction preserves finite length representations.

If the parabolic induction involves a representation which is irreducible cuspidal then the
corollary  follows from the above theorem. In general, any irreducible representation is a subrepresentation of a parabolically induced representation
of an irreducible cuspidal representation. We conclude the above corollary
by the transitivity of parabolic induction.

In order to show the analogous result for parabolic restriction, we need to use the following theorem due to
Howe whose proof for covering groups  is the same as in the linear case:

A smooth representation of $\wG$ is of finite length if and only if it is admissible and finitely generated.

A crucial tool that goes into the proof of Howe's theorem is Jacquet's lemma. Later, in <ref>, we will state a generalized version of it since it is necessary for the second adjointness.
Until then, let us give the classical statement

Let $V$ be an admissible representation of $\wG$, $\wK\le \wG$ an open compact subgroup admitting an Iwahori decomposition with respect to $\wP=\wL N$.
Then the following canonical map is surjective

\[V^{\wK}\to (V_N)^{\wK\cap \wL}.\]

Immediately from Jacquet's lemma we obtain

Parabolic restriction preserves admissibility.

Use Howe's theorem, the above corollary and the easy fact that parabolic restriction preserves finite type to deduce

Parabolic restriction preserves finite length representations.

§ THE BERNSTEIN DECOMPOSITION

We give an exposition of a small part of a theory due to Bernstein which
allows one to decompose the category $\cM(\wG)$ of smooth complex
representations of $\wG$  as a direct product of certain indecomposable full subcategories,
now called the Bernstein components of $\cM(\wG)$.
The results are due to [6] where it is also stated that they hold for finite central extensions of reductive $p$-adic groups.
Indeed, no essential modifications are needed to adapt the proof from [6] to the case of finite central extensions.
However, in the following we follow mostly the exposition of the linear case from <cit.>  giving
precise references in appropriate places.

The idea is that whereas $\wG$ does not have compact representations, the group $\wGo$ does,  and moreover, from Harish-Chandra's <ref>, all irreducible cuspidal representations of $\wG$ restrict to compact representations of $\wGo$.
So using the results of <ref>, we can decompose $\cM(\wG)$ into a cuspidal part and an induced part.
Using induction we express the induced part in a similar way.

§.§ Cuspidals split

In this section we sketch the decomposition of $\cM(\wG)$ into a product of cuspidal and induced.

We define $\cM(\wG)_\cs$ (resp., $\cM(\wG)_\indu$) to be the full subcategory of $\cM(\wG)$ formed of representations all of whose irreducible subquotients are cuspidal (resp., that have no cuspidal irreducible subquotients).
The set $\Irr_\cs(\wG)$ denotes the set of irreducible cuspidal representations of $\wG$.

The category $\cM(\wG)_\cs$  (resp., $\cM(\wG)_\indu$) is the pullback of the subcategory of compact representations (resp., non-compact representations) from $\cM(\wGo)$ through the functor $\Res_\wGo^\wG$.

Let $(V,\pi)\in\Irr_\cs(\wG)$ and denote by $[\pi]$ its inertia class, i.e., its orbit under $\cX(\wG)$.
From <ref> we have that the restriction of $\pi$ to $\wGo$ depends only on the inertia class and we know moreover that this restriction has finite length and is semisimple.

We define $\cM(\wG)_{[\pi]}$ to be the full subcategory of $\cM(\wG)$ formed of those representations
all of whose subquotients belong to $[\pi]$.
Similarly define $\cM(\wG)_{[\text{\rm out }\pi]}$.

Here is a different way of thinking about $\cM(\wG)_{[\pi]}$.
If $\tau_1,\tau_2,\dots,\tau_l$ are the irreducible summands of $\pi$ restricted to $\wGo$, then  the category $\cM(\wG)_{[\pi]}$ consists precisely in the representations of $\wG$ whose restriction to $\wGo$ are direct sums of $\tau_i, i=1,\dots, l$,
i.e.,  those
representations of $\wG$ whose restriction to $\wGo$ belongs to $\cM(\wGo)_{[\cA]}$, where $\cA = \{\tau_i\mid i=1,\dots, l\}$. (See <ref>.)

Harish-Chandra's <ref> together with the results from <ref> give

For an irreducible cuspidal representation $\pi$ of $\wG$,
we have a decomposition of the category $\cM(\wG)$ as

\[ \cM(\wG)\simeq \cM(\wG)_{[\pi]}\times \cM(\wG)_{[\text{out }\pi]}. \]

One uses the uniform admissibility <ref> to check the condition (<ref>) in <ref> and then applies <ref> and <ref> to get the following decomposition

The subcategories $\cM(\wG)_\cs$ and $\cM(\wG)_\indu$ split the category $\cM(\wG)$:

\begin{align}\label{Eq:dec cuspidal x induced}
\cM(\wG) = \cM(\wG)_\cs\times \cM(\wG)_\indu
\end{align}

and moreover the cuspidal part decomposes as

\begin{align}\label{Eq:dec cuspidals}
\cM(\wG)_\cs = \prod_{[\pi]\in[\Irr(\wG)_\cs]} \cM(\wG)_{[\pi]}.
\end{align}

Since the center of a product of categories is the product of their centers (see <ref>), we have

\begin{align*}
\cZ(\cM(\wG)) = \prod_{[\pi]\in[\Irr_\cs(\wG)]} \cZ_{[\pi]}\times \cZ(\cM(\wG)_\indu).
\end{align*}

It turns out that for an irreducible cuspidal representation $\pi$ the center $\cZ_{[\pi]}$ of $\cM(\wG)_{[\pi]}$ is not very hard to determine.
Recall the notation $\Lambda(\wG) = \wG/\wGo$ and $\cX(\wG) = \Hom_\gr(\Lambda(\wG),\bC^\times)$ and notice that the group algebra $\bC[\Lambda(\wG)]$ is the algebra of regular functions on the algebraic variety (a torus) $\cX(\wG)$.

We first need a lemma whose easy proof follows by  considerations of  the central character.
Given $\pi\in\Irr_\cs(\wG)$,
its stabilizer in $\cX(\wG)$,

\[ \cG_\pi:=\Stab_{\cX(\wG)}(\pi) = \left \{\chi \in \cX(\wG)\, \,  \left | \pi \otimes \chi \cong \pi \right .  \right \},\]

is finite.

Given $\pi\in\Irr_\cs(\wG)$, we have a canonical isomorphism

\begin{align*}
\cZ(\cM(\wG)_{[\pi]})\simeq \bC[\Lambda(\wG)]^{\cG_\pi} = \cO(\cX(\wG)/\cG_\pi).
\end{align*}

In particular, $\cZ(\cM(\wG)_{[\pi]})$ is isomorphic to a ring of Laurent polynomials and hence is smooth of Krull dimension equal to the the rank of $\Lambda(\wG)$.

Actually, one can do a bit better and find an equivalence of categories $\cM(\wG)_{[\pi]}\simeq \rmod\cR_{[\pi]}$ with $\cR_{[\pi]} := \End_{\wG}(\Pi_{[\pi]})$ and $\Pi_{[\pi]} := \ind_{\wGo}^\wG (\Res_{\wGo}^\wG(\pi))$.
This will be discussed in <ref>.

§.§ Induced representations

In this section we look at the category $\cM(\wG)_\indu$ and decompose it into blocks.
The main input is the geometric lemma, see  (<cit.>),   <ref> and <ref> above,
as well as
<cit.> for a proof that works also for $\wG$.

* A cuspidal datum is a couple $(\wL,\rho)$ where $\wL$ is a Levi subgroup of $\wG$ and $\rho\in\Irr(\wL)_\cs$ is an irreducible cuspidal representation of $\wL$.

* We say that two cuspidal data $(\wL,\rho), (\wM,\tau)$ are conjugate (or associate) if there exists $g\in\wG$ such that

\begin{align*}
\wL = {}^g\wM  \text{ and } \rho = {}^g\tau,
\end{align*}

* We say that two cuspidal data $(\wL,\rho), (\wM,\tau)$ define the same inertial support if there exists $g\in\wG$ and $\chi\in\cX(\wL)$ such that

\begin{align*}
\wL = {}^g\wM  \text{ and } \rho = {}^g\tau \chi.
\end{align*}

We denote by $\Omega(\wG)$ the set of cuspidal data up to conjugation and by $\cB(\wG)$ the cuspidal data up to conjugation and inertia.

<ref> guarantees that the following notion is well defined.

Let $\pi\in\cM(\wG)$ be an irreducible representation.
We define the cuspidal support of $\pi$ to be a cuspidal datum $(\wL,\rho)\in\Omega(\wG)$ (well-defined
up to conjugation in $\wG$)
such that $\pi$ appears in the Jordan–Hölder series of the parabolic induction $\bfi_{\wL,\wP}^\wG(\rho)$ for some parabolic subgroup $\wP$ with Levi decomposition $\wP=\wL N$.

In order to decompose $\cM(\wG)_\indu$ we would like to pull back, for each cuspidal datum up to conjugation $(\wL,\rho)\in\Omega(\wG)$, the decomposition from (<ref>) for $\wL$ to $\wG$ for each irreducible cuspidal representation of $\wL$.
However, we need to take into account also the inertia because $\cM(\wL)_{[\rho]}$ is an indecomposable subcategory.

Recall the set $\cB(\wG)$ of cuspidal data up to conjugation and inertia.
For $(\wL,\rho)$ a cuspidal datum we denote by $[\wL,\rho]_\wG$ its class in $\cB(\wG)$.
If no confusion can arise, we drop the subscript $\wG$.

Given $[\wL,\rho]\in\cB(\wG)$, <ref> allows us to define unambiguously the subcategory of representations of $\wG$ all whose irreducible subquotients have cuspidal support in $[\wL,\rho]$:

\begin{align}
\cM(\wG)_{[\wL,\rho]} = \left\{\pi\in \cM(\wG)\bigg| \begin{array}{l}\text{ all irreducible subquotients of $\pi$ as a $\wG$-module}\\
\text{ have cuspidal support  in } [\wL,\rho]\end{array}\right\}.
\end{align}

The next lemma gives another characterization of the category $\cM(\wG)_{[\wL,\rho]}$.

Let $[\wL,\rho]\in\cB(\wG)$ and $\wP=\wL N$ be a parabolic subgroup with Levi $\wL$.
Then the subcategory $\cM(\wG)_{[\wL,\rho]}$ is the smallest full subcategory of $\cM(\wG)$ closed under subquotients and containing $\bfi_{\wL,\wP}^\wG(\cM(\wL)_{[\rho]})$.

It follows easily from <ref> and <ref>.

Use Frobenius reciprocity, <ref> and exactness of parabolic induction and restriction to deduce the following proposition.

If $[\wL,\rho], [\wL',\rho']$ are distinct elements of $\cB(\wG)$,  then the subcategories $\cM(\wG)_{[\wL,\rho]}$ and $\cM(\wG)_{[\wL',\rho']}$ are (derived) orthogonal, i.e., there are no non-zero Ext groups between them.

The above proposition together with <ref> imply that we have a fully faithful functor

\[ \prod_{\fs\in\cB(\wG)}\cM(\wG)_\fs \hookrightarrow \cM(\wG).\]

Proving that this functor is essentially surjective[A functor $ F\colon \cC\to \cD$ is called essentially surjective if for any $X\in \cD$ there exists $A\in\cC$ such that $F(A)\simeq X$.] gives the Bernstein decomposition for $\cM(\wG)$:

The category of smooth representations of $\wG$ decomposes into blocks indexed by $\cB(\wG)$:

\begin{equation}\label{Eq:Bernstein decomposition}
\cM(\wG) = \prod_{{\fs} \in \mathcal{B}(\wG)} \cM(\wG)_\fs.
\end{equation}

Using <ref> what is left to show is that every representation $\pi\in\cM(\wG)_\indu$ can be written as a direct sum of representations $\pi_\fs\in\cM(\wG)_\fs$:

\begin{align}\label{Eq:decompose rep along subcategories indexed by B(G)}
\pi \simeq \oplus_{\fs\in\cB(\wG)}\pi_\fs \quad \text{ for some  } \pi_\fs\in\cM(\wG)_\fs.
\end{align}

First one observes that (<ref>) holds  for an induced representation of a cuspidal representation from a Levi.
This is tautological using <ref>.

Next one notices that if $\pi'\subset \pi$ is a subrepresentation and $\pi$ satisfies (<ref>) then the same is true of $\pi'$.
Namely, consider $\pi'_\fs=  \pi_\fs \cap \pi' $, then	$\pi'= \oplus_{\fs\in\cB(\wG)}\pi'_\fs$ as follows easily only using the fact that no nonzero subquotients of  $\pi_\fs$ for different $\fs$ are isomorphic (analogous assertion in group theory is called Goursat's Lemma).

Finally, one needs to show that every representation $\pi\in\cM(\wG)_\indu$ can be embedded into a sum of induced representations.
This is done using a pair of adjoint functors that are direct sums of all parabolic induction functors (resp., parabolic restriction) indexed by all the standard parabolic subgroups.
For details, see <cit.>.

As a consequence one can show that $\cM(\wG)$ is a noetherian category and this allows one to show that parabolic induction preserves finite type representations.
Below we sketch the strategy.

Recall that in an abelian category $\cA$ an object $X$ is said to be

* noetherian if all increasing sequences of subobjects of $X$ are stationary.

* of finite type if whenever $X=\cup_i X_i$ with $X_i\subset X$,  there exist finitely many indices $i_1,\dots,i_n$ such that $X=\cup_{k=1}^n X_{i_k}$.

An abelian category is called noetherian if all its finite type objects are noetherian.

Thanks to Bernstein's decomposition theorem and the fact that each cuspidal block is equivalent to modules over a noetherian algebra (see <ref>) one can show:

The category of smooth representation $\cM(\wG)$ is noetherian.

Using the above theorem and the simple fact that parabolic restriction preserves finite type (see the discussion in <ref>) one shows:
Parabolic induction preserves finite type representations.

This theorem is used in <ref> to show that the generator $\Pi_{\fs}$ of a non-cuspidal block is of finite type.

§ THE SECOND ADJOINTNESS THEOREM

Recall that the Frobenius reciprocity theorem asserts that for $\wP$ a parabolic subgroup of $\wG$ with Levi subgroup
$\wL$, the functor $\bfr_{\wL,\wP}^\wG$ is left adjoint to $\bfi_{\wL,\wP}^\wG$.
It is natural to look for an adjoint in the other direction.

Although showing that $\bfi_{\wL,\wP}^\wG$ admits a right adjoint is not difficult (see below), identifying
this adjoint precisely is quite hard and is the content of Bernstein's second adjointness theorem: $\bfi_{\wL,\wP}^\wG$ is left adjoint to $\bfr_{\wL,\wP^-}^\wG$, where $\wP^-$ is the opposite parabolic of $\wP$.

In this section we will sketch the main steps in the proof following the exposition in <cit.> where all the details are spelled out for linear groups.
In loc.cit., the argument is streamlined using the notion of completion of a representations of $\wG$ which we recall below.
Other references for the second adjointness theorem for the linear case include <cit.>,
[10]; a geometric proof is given in [16].

§.§ Completion

We recall the notion of completion for smooth $G$-modules where $G$ is a locally compact, totally disconnected group admitting a countable basis of neighborhoods of 1 given by compact open subgroups.
For $V$ a smooth $G$-module its completion $\overline{V}$ is a $G$-module (not smooth anymore)
sitting between $V$ and $(V^{\vee})^*$.

This notion will play a simplifying role in the proof of the second adjointness theorem and it also helps clarify the relationship between $N$-invariants and $N$-coinvariants.

Notice that for a representation $V$ of $G$ and compact open subgroups $L\subset K$ with corresponding idempotents $e_K,e_L\in\cH(G)$,  we have a natural map $e_L V\to e_K V$ sending $w$ to $e_Kw$.

For $V$, a smooth representation of $G$, we define the functor

\[ \overline{(\,\,)}\colon \cM(G)\to \Rep_\bC(G) \text{ by }\]

\[ \overline{V}:=\varprojlim (e_K V) = \varprojlim (V^K)\]

where the inverse limit is taken over all compact open subgroups of $G$.

A few remarks are in order.

* Since the set of compact open subgroups of $G$ is invariant under conjugation by $G$, it follows that  $\overline{V}$is a $G$-module and that $\overline{V}^K=V^K$.
Clearly $V\subset  \overline{V}$ and the smooth part of  $\overline{V}$ is $V$.

* The completion functor is exact because the Mittag-Leffler condition is automatically satisfied.

* Another way of writing the completion functor is to note that
acquires a $G$-module structure since $\cH(G)$ is a $G \times G$-module,
and since $\cH(G)=\varinjlim \cH(G)e_K$, it follows that

\[ \overline{V} = \Hom_G(\cH(G),V). \]

Note that the exactness of the functor $V \rightarrow \overline{V}$
is equivalent to $\cH(G)$ being a projective $G$-module.

For $V$ a smooth representation of $G$, we have

\[ \overline{V^\vee} = V^*.\]

Suppose given $B\colon V\times W\to \bC$ a pairing of smooth $G$-representations inducing an injective map of representation $V\hookrightarrow W^*$.
Since $(W^*)^K = (W^K)^*$ for any smooth representation $W$ of $G$ and any compact open subgroup $K \subset G$, we obtain

\[ \overline{V} = \varprojlim (V^K) \subset \varprojlim(W^K)^* = (\varinjlim W^K)^* = W^*.\]

In particular, taking $B\colon V^\vee\times V\to \bC$ to be
the natural pairing and using that $(V^\vee)^K = (V^*)^K$ for all open compact subgroups $K$,  we obtain the claimed equality.

Applying it to the standard bilinear form $B\colon \cH(G) \times \cH(G) \to \bC,$ we find

The completion $\overline{\cH}(G)$ of $\cH(G)$ is the space of distributions $D$ on $G$ such that $e_K*D$ is a compactly supported distribution on $G$ for all idempotents $e_K$.

In particular, coupled with <ref>,  we deduce:

The center of $\cM(G)$ can be described as $\overline{\cH}(G)^G$, i.e.,  the space of invariant distributions $D$ on $G$ such that  $e_K*D$ is a compactly supported distribution on $G$ for all idempotents $e_K$.

It is known that for smooth representations taking invariants with respect to a unipotent subgroup $N$ is not a well behaved functor (for example, it is not exact) and as such it is almost never used.
On the other hand, taking coinvariants under $N$ leads to nice exact functors (Jacquet functors) which are extremely important in the theory.
The completion allows us to reconcile invariants and coinvariants for $N$ as shown below.

The functor $(-)^N\circ \overline{(-)} \circ (-)^\vee\colon \cM(G)\to\Vect_\bC$ is exact.

Notice that we have a natural isomorphism $(V^*)^N = (V_N)^*$ for any representation $V$ of $G$.
In particular, for a smooth representation $V$ of $G$, using <ref> we obtain

\[  \overline{V^\vee}^N = (V^*)^N = (V_N)^*.\]

Taking $N$-coinvariants is an exact functor because $N$ is an increasing union of compact open subgroups.
So the functor $(-)^N\circ \overline{(-)} \circ (-)^\vee$ identifies with the composition of two exact functors $(-)^*\circ (-)_N$ which concludes the proof.

In other words, taking $N$-invariants on completions of contragredient $G$-modules is an exact functor.
In particular, we deduce

The functor $(-)^N\circ \overline{(-)}$ is exact on admissible modules.

§.§ Adjoints

From now on, $G$ is a reductive $p$-adic group and $\wG$ is a finite covering group of it (see <ref>).
We fix $\wP = \wL N$, a parabolic subgroup with a Levi decomposition in $\wG$, and we denote by $\wP^-$, the opposite parabolic subgroup with Levi decomposition $\wL N^-$.
We denote by $\delta_\wP$ the modulus character of $\wP$.

The induction functor $\ind_\wP^\wG \colon \cM(\wP)\to \cM(\wG)$ has a different, more algebraic, expression in terms of the Hecke algebras.
First, recall that we have an equivalence of categories

\[ \cM(\wG)\simeq \cH(\wG)\lmod\nd \]

between smooth representations of $\wG$ and non-degenerate $\cH(\wG)$-modules and that this holds also for $\wP$ and $\wL$.

The following result gives a different incarnation of the induction from $\wP$ to $\wG$:

There is a natural isomorphism of functors

\[ \cH(\wG)\otimes_{\cH(\wP)} - \simeq \ind_\wP^\wG \circ (-\otimes_\bC \delta_\wP). \]

Now consider the functor of taking $\wG$-smooth vectors (or the non-degenerate submodule for the Hecke algebra)

\[ (-)_{nd\mhyphen\wG}\colon \cH(\wG)\lmod \to \cH(\wG)\lmod\nd\]

which to a module associates its non-degenerate submodule.
Similarly for $\cH(\wP)$-modules.

The functor $(-)_{nd\mhyphen\wG}$ is right adjoint to the inclusion functor
\[\cH(\wG)\lmod\nd\hookrightarrow \cH(\wG)\lmod. \]

The functor

\[ \ind_\wP^\wG \circ (-\otimes_\bC \delta_\wP)\colon \cM(\wP)\to\cM(\wG) \]

is left adjoint to

\[ (-)_{nd-\wP}\circ \Res_\wP^\wG\circ \overline{(-)}\colon\cM(\wG)\to \cM(\wP).\]

The usual tensor–Hom adjunction has the following analogue here:
the functor

\[ \cH(\wG)\otimes_{\cH(\wP)} -\colon \cM(\wP)\to\cM(\wG) \]

is left adjoint to

\[ (-)_{nd\mhyphen \wP}\circ \Hom_{\cH(\wG)}(\cH(\wG),-)\colon \cM(\wG)\to\cM(\wP) \]

where $\cH(\wG)$ is viewed as a left $\cH(\wG)$-module and a right $\cH(\wP)$-module.