Harmonic analysis on $\mathsf{GL}_n$ over finite fields

There are many formulas that express interesting properties of a finite group $G$ in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio\[\frac{\operatorname{trace}(\pi (g))}{\operatorname{dim}(\pi)} \: \textrm{,}\]for an irreducible representation $\pi$ of $G$ and an element $g$ of $G$. For example, in [12] the authors stated a formula of this type for analyzing certain random walks on $G$.

It turns out [22, 23] that for classical groups $G$ over finite fields (which provide most examples of finite simple groups) there are several (compatible) invariants of representations that provide strong information on the character ratios. We call these invariants collectively rank.

Rank suggests a new way to organize the representations of classical groups over finite and local fields—a way in which the building blocks are the “smallest” representations. This is in contrast to Harish–Chandra’s philosophy of cusp forms that is the main organizational principle since the 60s, and in it the building blocks are the cuspidal representations which are, in some sense, among the “largest.” The philosophy of cusp forms is well adapted to establishing the Plancherel formula for reductive groups over local fields, and led to Lusztig’s classification of the irreducible representations of such groups over finite fields. However, analysis of character ratios seems to benefit from a different approach.

In this note we discuss further the notion of tensor rank for $\mathsf{GL}_n$ over a finite field $\mathbb{F}_q$ and demonstrate how to get information on representations of a given tensor rank using tools coming from the recently studied eta correspondence, as well as the well known philosophy of cusp forms, mentioned just above.

A significant discovery so far is that although the dimensions of the irreducible representations of a given tensor rank vary by quite a lot (they can differ by large powers of $q$), for certain group elements of interest the character ratios of these irreps are nearly equal to each other. Thus, for purposes of this aspect of harmonic analysis, representations of a fixed tensor rank form a natural family to study.

For clarity of exposition, we illustrate the developments with the aid of a specific motivational example that shows how one might apply the results to certain random walks.

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Dedicated to the memory of Bertram Kostant.

Received 19 December 2019

Accepted 12 August 2020

Published 22 December 2021