Hopf algebra and the duality operation for $\mathfrak{gl}_n(\mathbb{F}_q)$

TL;DR

This paper explores the structure of invariant functions on the Lie algebra of general linear groups over finite fields using Hopf algebra theory, revealing new duality insights and connections to Harish-Chandra induction.

Contribution

It introduces a real PSH algebra framework for invariant functions and characterizes the duality operation for bla_n(\u00bb_q), providing new proofs of classical results.

Findings

Invariant functions are linear combinations of Harish-Chandra inductions.

Two specific bla(a0_q)-lattices form real PSH algebras.

A new conceptual characterization of the duality operation.

Abstract

In this paper we study the space $C(\mathfrak{gl}_n(\mathbb{F}_q))$ of complex invariant functions on $\mathfrak{gl}_n(\mathbb{F}_q)$, through a Hopf algebra viewpoint. First, we consider a variant notion of Zelevinsky's PSH algebra defined over the real numbers $\mathbb{R}$. In particular, we show that two specific $\mathbb{R}$-lattices inside the complex Hopf algebra $\bigoplus_nC(\mathfrak{gl}_n(\mathbb{F}_q))$ are real PSH algebras, and that they do not descend to $\mathbb{Z}$. Then, among consequences, we prove that every element in $C(\mathfrak{gl}_n(\mathbb{F}_q))$ is a linear combination of Harish-Chandra inductions of Kawanaka's pre-cuspidal functions, and give a conceptual characterisation of duality operation for $\mathfrak{gl}_n(\mathbb{F}_q)$, which in turn allows us to give a new proof of a classical result of Kawanaka.