Invariant functor for admissible representations of reductive groups over local fields

Question

Hello,

I have a question concerning a certain functor between represention categories. I'm rather sure this is already known, but I could not find a reference.

Let $F$ be a local non-archimedean field with ring of integers $\mathcal{O}$, maximal ideal $\wp\subset\mathcal{O}$ and finite residue field $\mathbb{F}_q$. Denote by $Rep(GL(2,F))$ the category of isomorphism classes of admissible representations of $GL(2,F)$ on a complex vector space. Then there is a functor to the category $Rep(GL(2,\mathbb{F}_q))$ of representations of $GL(2,\mathbb{F}_q)$ defined by $\rho\mapsto(res_{GL(2,\mathcal{O})}(\rho))^{\Gamma(\wp)}$. That means, a representation $\rho$ first gets restricted to $GL(2,\mathcal{O})$ and then one applies the invariant functor to obtain the space of vectors fixed by $\Gamma(\wp)=ker(GL(2,\mathcal{O})\to GL(2,\mathbb{F}_q))$. It should be possible to replace $G$ by an arbitrary connected affine reductive algebraic group with a corresponding $\mathcal{O}$ group scheme.

For lack of a better name, let me call this $\mathcal{F}:Rep(GL(2,F)\to Rep(GL(2,\mathbb{F}_q))$. This functor should be well known and its properties have already been studied by Bernstein, for example in "Le "centre" du Bernstein".

My first question is: Is there already a standard notation for what I call $\mathcal{F}$?

$\mathcal{F}$ has some very nice properties: It is exact, it maps supercuspidal representations to cuspidal ones and it commutes with parabolic induction.

I calculated some examples for $GL(2)$ and $GSp(4)$ and it showed that if $\rho\in Rep(GL(2,F))$ is irreducible and generic (i.e. has a Whittaker model), then $\mathcal{F}(\rho)\in Rep(GL(2,\mathbb{F}_q))$ is either zero or also generic. This holds for most of the non-supercuspidal representations of $GSp(4,F)$, too.

My second question is: Does this hold for arbitrary affine reductive groups? Could somebody please point me to a reference, where this is worked out?

Kind regards, Mirko

Answer 1

Here is a possible general context for such functors.

Take $G$ a $p$-adic reductive group and $X$ its extended Bruhat-Tits building. For a point $x\in X$, one has the fixator $G_x$, the pro-$p$-radical $G_x^+$ of $G_x$ and the "reductive" quotient $\bar G_x$. Hence you get a functor $V\mapsto V^{G_x^+}$ from $Rep(G)$ to $Rep(\bar G_x)$.

This functor has the following nice property : let $T$ be a torus whose associated appartment contains $x$ and let $P$ be a psgp of $G$ that contains $T$. Denote by $U$ the unipotent radical of $P$ and by $M$ the Levi component of $P$ that contains $T$. Then the image $\bar P_x$ of $P\cap G_x$ in $\bar G_x$ is a psgp of $\bar G_x$ with Levi decomposition $\bar M_x \bar U_x$ (with obvious notation). In this context we have functorial isomorphisms
$$ (V_U)^{M_x^+}\simeq (V^{G_x^+})_{\bar U_x}. $$

This explains why the functor takes cuspidals to cuspidals. Also, taking adjoint, you get an isomorphism between induced representations (mixing parabolic induction and compact induction).

Note that taking $G_x^+$-invariants is the same as applying a certain idempotent to $V$. There is a generalisation for deeper level representations where one applies idempotents associated to Bushnell-Kutzko simple characters. This is explained in this paper .

But you raise an interesting question regarding the compatibility with "genericity" in the Whittaker sense. I don't know if this has been studied in this general context.

Answer 2

I can give only an answer to the 2nd question:

The functor $\mathcal{F}$ is a composition of restriction to $G(o)$ and then projection onto the subrepresentation of $G(o)$, which have kernel $\bmod p$.

Both of these functors are very wellbehaved by the Peter Weyl Theorem.

So all the properties you want follow in the case of a reductive group over a local non archimedean field, but it seems to me that you will study only very few representation with this (finitely many?).

Edit: The observation with cuspidal mapsto supercuspidal and parabolic induced to parabolic induced is best explained with the restriction induction formula for composing restriction and induction, perhaps modulo the assumption that supercuspidal are induced from maximal compacts (I do not know, if this is true in the generality you want this.)