Invariant functor for admissible representations of reductive groups over local fields

2011-08-30T09:24:17Z

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

metric on the space of real analytic functions

2011-01-27T23:42:17Z

Hello, this question may be simple but I couldn't find a reference.

Let $E$,$F$ be real Banach spaces and $\Omega\subset E$ be a bounded domain and let $C_b^{\omega}(\Omega,F)$ be the vector space of bounded real analytic functions from $\Omega$ to $F$. Now I would like to know if there is a natural way to define a metric on $C_b^\omega(\Omega,F)$, that makes the space complete.

Concretely, I have a series of real analytic functions that converge uniformly as well as their Frechet derivatives and now I would like to know if their limit is analytic again.

My first idea was to show that $C_b^\omega(\Omega,F)$ is a closed subspace of $C_b^\infty(\Omega,F)$. Here $C_{b}^{\infty}(\Omega,F)$ is the set of infinitely Frechet-differentiable functions equipped with the usual set of seminorms (i.e. $\Vert f\Vert_k:=\Vert D^k f\Vert_\infty$) defining a Frechet space and thus a metric $d(f,g):=\sum_{k=1}^\infty 2^{-k}\frac{\Vert f-g\Vert_k}{1+\Vert f-g\Vert_k}$ which makes $C_{b}^{\infty}(\Omega,F)$ a complete metric space.

Actually, I have no idea if this is true at all. Could someone please confirm if this is the right thing to prove or not?

Regards, Mirko

User mirko - MathOverflow

Invariant functor for admissible representations of reductive groups over local fields

metric on the space of real analytic functions