On a Bruhat decomposition related to the Shalika subgroup of $GL(2n)$

TL;DR

This paper explicitly describes the double coset structure of the group $GL(2n)$ over a field $F$ with respect to the Shalika subgroup and a maximal parabolic subgroup, providing a Bruhat decomposition relevant for representation theory.

Contribution

It provides a complete set of double coset representatives and a Bruhat decomposition for $GL(2n)$ related to the Shalika subgroup and maximal parabolics, extending understanding in representation theory contexts.

Findings

Explicit double coset representatives are obtained.

Cardinality of the double coset space is computed.

A Bruhat decomposition is established for certain parabolics.

Abstract

Let $F$ be a non-archimedean local field or a finite field. In this article, we obtain an explicit and complete set of double coset representatives for $S\backslash GL_{2n}(F)/Q$ where $S$ is the Shalika subgroup and $Q$ a maximal parabolic subgroup of the group $GL_{2n}(F)$ of invertible $2n\times 2n$ matrices. We compute the cardinality of $S\backslash GL_{2n}(F)/Q$ and also give an alternate perspective on the double cosets arising intrinsically from certain subgroups which are relevant for applications in representation theory. Finally, if $Q$ is a maximal parabolic subgroup of the type $(r,2n-r),$ we prove that $S\backslash GL_{2n}(F)/Q$ is in one to one correspondence with $\Delta S_n\backslash S_{2n}/S_{r}\times S_{2n-r}$ leading to a Bruhat decomposition. The results and proofs discussed in this article are valid over any arbitrary field $F$ even though our motivation is from representation theory of $p$-adic and finite linear groups.