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