A Look at Representations of SL(2,q) Via the Lens of Size - Motivations, Rank, Eta Correspondence

Explore a groundbreaking approach to harmonic analysis on finite groups through this 41-minute conference talk that examines representations of SL(2,q) using innovative concepts of "size" and "rank." Learn how traditional harmonic analysis principles—which decompose functions on the real line into frequency components—can be extended to finite classical groups through work developed with Roger Howe. Discover how class functions on finite groups can be expressed as linear combinations of irreducible characters, with a newly defined notion of "frequency" or "size" called rank that enables systematic analysis of these mathematical objects. Understand the practical implications of this theoretical framework, including how it provides the first clear methods for bounding values of irreducible characters on various group elements—a problem that previously lacked adequate analytical tools. Follow the speaker's detailed examination of SL(2,q), the group of 2×2 matrices with entries in finite field Fq and determinant one, as the first nontrivial example demonstrating this revolutionary approach to group theory and representation theory.