One algebra of double cosets for a general linear group over a finite field

TL;DR

This paper studies an algebraic structure related to double cosets in general linear groups over finite fields, providing generators, relations, and an interpolation to complex parameters.

Contribution

It describes the algebra of bi-invariant functions on certain linear groups and extends it to complex dimensions, offering new algebraic insights.

Findings

Explicit generators and relations for the algebra

Interpolation of the algebra to complex values of n

Structural understanding of double coset algebras

Abstract

Let $\mathbb {F}_q$ be finite field with $q$ elements. Let $\alpha\leqslant n$ be positive integers. Consider the general linear group $\mathrm{GL}(\alpha+n, \mathbb {F}_q) $ and its subgroup $H(n)$, which fixes the first $\alpha$ basis elements in $\mathbb {F}_q^{\alpha+n}$. Denote $\mathcal{A}_n$ by the convolution algebra of $H(n)$-biinvariant functions on $\mathrm{GL}(\alpha+n, \mathbb {F}_q) $. We describe algebras $\mathcal{A}_n$ in terms of generators and relations and show that the family $\mathcal{A}_n$ admits a natural interpolation to arbitrary complex $n$ (the field $\mathbb {F}_q$ and $\alpha$ are fixed).