# Projectors separating spectra for $L^2$ on pseudounitary groups $U(p,q)$

## TL;DR

This paper explicitly constructs orthogonal projectors in the $L^2$ space on pseudo-unitary groups $U(p,q)$ to separate spectra into distinct types, addressing a classical question by Gelfand and Gindikin.

## Contribution

It provides explicit formulas for orthogonal projectors in $L^2(U(p,q))$ that isolate spectral components, including new finer spectral decompositions based on discrete series and tempered parameters.

## Key findings

- Explicit orthogonal projectors for spectral separation in $L^2(U(p,q))$
- New spectral decompositions based on discrete series representations
- Addresses classical spectral separation question by Gelfand and Gindikin

## Abstract

The spectrum of $L^2$ on a pseudo-unitary group $U(p,q)$ (we assume $p\ge q$ naturally splits into $q+1$ types. We write explicitly orthogonal projectors in $L^2$ to subspaces with uniform spectra (this is an old question formulated by Gelfand and Gindikin). We also write two finer separations of $L^2$. In the first case pieces are enumerated by $r=0$, 1,..., $q$ and representations of discrete series of $U(p-r,q-r)$, where $r=0$, \dots, $q$. In the second case pieces are enumerated by all discrete parameters of the tempered spectrum of $U(p,q)$.