On multiplication of double cosets for $\GL(\infty)$ over a finite field

TL;DR

This paper studies the algebraic structure of double cosets in an infinite general linear group over a finite field, revealing a semigroup structure and its action on fixed vectors in certain representations.

Contribution

It introduces a semigroup structure on double cosets of $GL()$ by a subgroup $P$, and explores its action on fixed vectors in unitary representations.

Findings

Double cosets form a natural semigroup.

The semigroup acts on $P$-fixed vectors in representations.

Provides a new algebraic framework for infinite linear groups over finite fields.

Abstract

We consider a group $GL(\infty)$ and its parabolic subgroup $B$ corresponding to partition $\infty=\infty+m+\infty$. Denote by $P$ the kernel of the natural homomorphism $B\to GL(m)$. We show that the space of double cosets of $GL(\infty)$ by $P$ admits a natural structure of a semigroup. In fact this semigroup acts in subspaces of $P$-fixed vectors of some unitary representations of $GL(\infty)$ over finite field.