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!



reference help about a result on representation theory
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.


