reference help about a result on representation theory

Question

I read the following theorem in a paper without a proof, which I don't understand well. Let $F$ be a global function field, and $v$ be a place of $F$, use $G_r$ to denote $GL_r$.

Theorem: For any integer $r\ge 1$, we have

(1) the space of unitary irreducible admissible representations $\pi$ of $G_r(F_v)$ can be decomposed as the disjoint union of real algebraic varieties $Im[\pi_0]/Fixe(\underline{r},\pi_0)$ indexed by the pair $(\underline{r},\pi_0)$ where

$\bullet$ $\underline{r}$ denotes a partition $r=r_1+...+r_k$ of $r$,

$\bullet$ $\pi_0$ is a unitary irreducible repersentation and square integrable on $G_{r_1}(F_v)\times...\times G_{r_k}(F_v)$,

$\bullet$ $[\pi_0]$ is a complex algebraic variety on which the complex torus $(\mathbb{C}^{\times})^k$ acts simply transitively,

$\bullet$ $Im[\pi_0]$ is a real compact subvariety of $[\pi_0]$ on which the subtorus $U(1)^k$ acts simply transitively,

$\bullet$ $Fixe(\underline{r},\pi_0)$ is a finite group which acts on both $[\pi_0]$ and $Im[\pi_0]$.

This allows us to talk about polynomial functions on this space: These are the functions zero outside a finite number of components whose restriction to each component of $Im[\pi_0]/Fixe(\underline{r},\pi_0)$ are polynomials which can be extended to polynomials on the complex variety $[\pi_0]$ invariant by $Fixe(\underline{r},\pi_0)$.

(2) This space is given the Plancherel measure, such that every compactly supported locally constant function

$$h_v(g):G_r(F_v)\to \mathbb{C}$$ can be spectrally decomposed in the form

$$h_v(g)=\int h_{v,\pi}(g)d\pi, \ \ \ \ \ \ g\in G_r(F_v)$$ where

$\bullet$ each $h_{v,\pi}:G_r(F_v)\to \mathbb{C}$ is an element in the eigenspace associated to $\pi$, that is, is the linear combination of matrix coefficients of $\pi$,

$\bullet$ each function $\pi\to h_{v,\pi}(g)$ is a polynomial.

My question is

(1) What is a reference for this theorem?

(2)This theorem is stated for global function field, and is an analog still true for local fields (archimedean or nonarchimedean) of a global number field? If so, is there a reference?

Thank you very much for any explanation or reference suggestion!

Answer 1

1. Every unitary $\infty$-dim'l irreducible representation can be writen as inducing a square-integrable representation from a parabolic subgroup with Levi subgroup $G' =G_{r_1} \times \dots G_{r_2}$. Googling for Bernstein center might help.

2. I guess $[\pi_0]$ is the family, where you tensor by unramified one-dimensional representations of $G'$, so that is not a fancy action but does not preserve unitarity.

3. The character formula for a prabolic induced rep looks for parabolic $P_r = G_r N_r$ and Iwasawa decomposition $G = P_r K_r$ like $$ tr\; \pi(\phi) = tr\; \pi(\phi^{N,K}) , \qquad \phi^{N,K}(g) := \int\limits_{N}\int\limits_{K} \phi(k^{-1}gnk) d k\; d n\;(g \in G_r).$$ This is an exercise in functional analysis.

4. I can not address the algebraic geometry stuff, but it is probably not that difficult. I would need a definition of $Im[\pi_0]$ and the finite group though, but I have no idea what that should be. May be the finite group means permuting $G_k$'s of equal dimension, that is, the relative Weyl group?

All what I am mentioning holds for arbitrary local fields, but the polynomial statement will only be true for non-archimedean fields.

I would check Laumon - Cohomology of Drinfeld modules I and probably II. Most likely, he does the local things without specifying the characteristic of the local field, but of course does not address the archimedean situation at all.