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

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 $G$.