Positivity of principal matrix coefficients of principal series representations of $GL_{n}(R)$

TL;DR

This paper investigates the positivity of matrix coefficients in principal series representations of the real general linear group, providing explicit descriptions of certain group decompositions and their implications.

Contribution

It determines the image of the exponential of the Cartan subspace under the Iwasawa projection and proves a positivity result for principal series matrix coefficients.

Findings

Explicit description of the Iwasawa projection of exp(p_0)

Positivity of principal series matrix coefficients established

Enhanced understanding of the structure of principal series representations

Abstract

Let $G=GL_{n}(R)$, with the usual Cartan decomposition $G=Kexp(p_0)$ and the usual Iwasawa decomposition $G=NAK$. We determine the image of $exp(p_0)$ under the projection of $G$ to $K$ through the Iwasawa decomposition. As an application, we prove a positivity result about the matrix coefficients of principal series representations of $G$.