On double cosets of groups $GL(n)$ with respect to subgroups of block strictly triangular matrices

TL;DR

This paper parametrizes double cosets of the general linear group with respect to certain block triangular subgroups and analyzes the associated quasi-regular representation, revealing its spectral properties and symmetries.

Contribution

It provides a new parametrization of double cosets for $GL(n)$ with respect to block triangular subgroups and studies the spectral and symmetry properties of the related representation.

Findings

Parametrization of double cosets of $GL(n)$ with respect to block triangular subgroups.

Identification of additional symmetries in the quasi-regular representation.

The spectrum of the representation is found to be multiplicity free.

Abstract

We parametrize the space of double cosets of the group $GL(n,\Bbbk)$ with respect to two subgroups $T_-$, $T_+$ of block strictly triangular matrices. In Addendum, we consider the quasi-regular representation of $GL(n,\Bbb{C})$ in $L^2$ on $T_- \ GL(n,\Bbb{C})$, observe that it admits an additional group of symmetries, find the joint spectrum, and observe that it is multiplicity free.