Temperedness criterion of the tensor product of parabolic induction for $GL_n$

TL;DR

This paper establishes a criterion for when tensor products of unitarily induced representations from parabolic subgroups of GL(n) are tempered, and provides L^p estimates for matrix coefficients of regular representations.

Contribution

It offers a necessary and sufficient condition for temperedness of tensor products of parabolic inductions in GL(n), and derives L^p bounds for matrix coefficients.

Findings

Characterization of tempered tensor products for GL(n)

L^p estimates for matrix coefficients on G/L

Conditions linking parabolic subgroups and temperedness

Abstract

We give a necessary and sufficient condition for a pair of parabolic subgroups $P$ and $Q$ of $G=GL_n(\mathbb{R})$ such that the tensor product of any two unitarily induced representations from $P$ and $Q$ are tempered. We also give an $L^p$-estimate of matrix coefficients of the regular representations on $L^2(G/L)$ when $L$ is a Levi subgroup of $G$.