A look at Representations of SL(2,F_q) through the Lens of Size

TL;DR

This paper explores the representations of the group SL(2,F_q) using harmonic analysis, revealing new insights into the role of certain representations previously considered anomalous, within a broader theoretical framework.

Contribution

It provides a new perspective on the representations of SL(2,F_q), connecting classical results with modern harmonic analysis techniques and reinterpreting previously considered anomalous representations.

Findings

Identifies the role of 'anomalous' representations as fundamental building blocks.

Develops a general theory for analyzing functions on finite classical groups.

Provides new insights into the structure of irreducible representations of SL(2,F_q).

Abstract

How to study a nice function on the real line? The physically motivated Fourier theory technique of harmonic analysis is to expand the function in the basis of exponentials and study the meaningful terms in the expansion. Now, suppose the function lives on a finite non-commutative group G, and is invariant under conjugation. There is a well-known analog of Fourier analysis, using the irreducible characters of G. This can be applied to many functions that express interesting properties of G. To study these functions one wants to know how the different characters contribute to the sum? In this note we describe the G=SL(2,F_q) case of the theory we have been developing in recent years which attempts to give a fairly general answer to the above question for finite classical groups. The irreducible representations of SL(2,F_q) are "well known" for a very long time [Frobenius1896, Jordan1907, Schur1907] and are a prototype example in many introductory courses on the subject. We are happy that we can say something new about them. In particular, it turns out that the representations that were considered as "anomalous" in the "old" point of view (known as the "philosophy of cusp forms") are the building blocks of the current approach.