4 deleted 355 characters in body; edited title; edited title

# How does the groupvNarightregular of $GL(n,\mathbb{R})$GL(n,R) and $GL(n,\mathbb{Q}_p)$GL(n,Qp) decomposeintoadirectintegraloffactors?

The question is contained in the title. I'd be glad, if I would guess that this question is already answered in the literature.

A little bit more demanding, and perhaps to general:

Given a the reductive group $G$ GL(n)$over a complete local field, how does the right regular representation on$L^2(G(F))$or perhaps better on$L^2(G(F)/Z(F))$or$L^2(G(F)^1)$decompose, where$Z$is the center of$G$. Even less intuitive, how does the right regular decomposition on$C_{comp}^{\infty} ( G(F) /Z(F) )$on the smooth, compactly supported function (C_{c}^{\infty}$-version or on the Bruhat-Schwartz algebra) Harish-Chandra Schwartz space look?Where are similar questions treated for Lie groups?

Note, that the Peter Weyl theorem for compact groups gives actually a decomposition as $G \times G$ module, with respect to the right and left representation, and such a decomposition is of course preferred also in this picture.

3 added 1 characters in body

The question is contained in the title. I'd be gladis , if this question is already answered in the literature.

A little bit more demanding, and perhaps to general: Given a reductive group $G$ over a local field, how does the right regular representation on $L^2(G(F))$ or perhaps better on $L^2(G(F)/Z(F))$ decompose, where $Z$ is the center of $G$. Even less intuitive, how does the right regular decomposition on $C_{comp}^{\infty} ( G(F) /Z(F) )$ on the smooth, compactly supported function (or on the Bruhat-Schwartz algebra) look? Where are similar questions treated for Lie groups?

Note, that the Peter Weyl theorem for compact groups gives actually a decomposition as $G \times G$ module, with respect to the right and left representation, and such a decomposition is of course preferred also in this picture.

2 added 18 characters in body

The question is contained in the title. I'd be glad is this question is already answered in the literature.

A little bit more demanding, and perhaps to general: Given a reductive group $G$ over a local field, how does the right regular representation on $L^2(G(F))$ or perhaps better on $L^2(G(F)/Z(F))$ decompose, where $Z$ is the center of $G$. Even less intuitive, how does the right regular decomposition on $C_{comp}^{\infty} ( G(F) /Z(F) )$ on the smooth, compactly supported function (or on the Bruhat-Schwartz algebra) look? Similar Where are similar questions treated for Lie groups?

Note, that the Peter Weyl theorem for compact groups gives actually a decomposition as $G \times G$ module, with respect to the right and left representation, and such a decomposition is of course preferred also in this picture.

1