Small Representations of Finite Classical Groups

TL;DR

This paper introduces a new framework for understanding small representations of finite classical groups, focusing on their rank and providing a method to construct and estimate their dimensions, with detailed analysis for symplectic groups.

Contribution

It develops a language centered on the notion of rank to describe small representations and proposes the eta correspondence as a method to construct all such representations.

Findings

Introduces the concept of rank for small representations.

Proposes the eta correspondence for constructing small representations.

Provides estimates on the dimensions of small representations based on rank.

Abstract

Finite group theorists have established many formulas that express interesting properties of a finite group in terms of sums of characters of the group. An obstacle to applying these formulas is lack of control over the dimensions of representations of the group. In particular, the representations of small dimensions tend to contribute the largest terms to these sums, so a systematic knowledge of these small representations could lead to proofs of important conjectures which are currently out of reach. Despite the classification by Lusztig of the irreducible representations of finite groups of Lie type, it seems that this aspect remains obscure. In this note we develop a language which seems to be adequate for the description of the "small" representations of finite classical groups and puts in the forefront the notion of rank of a representation. We describe a method, the "eta correspondence", to construct small representations, and we conjecture that our construction is exhaustive. We also give a strong estimate on the dimension of small representations in terms of their rank. For the sake of clarity, in this note we describe in detail only the case of the finite symplectic groups.