Suppose you have a finite group G and you want to study certain related structures (e.g., random walks, Cayley graphs, word maps, etc.). In many cases, this might be done using sums over the characters of G. A serious obstacle in applying these Fourier type formulas is lack of knowledge on the low dimensional representations of G. In fact, numerics shows that the "small" representations tend to contribute the largest terms to these sums, so a systematic knowledge on them might assist in the solution of important problems.

In this talk I will discuss a joint project (see arXiv:1609.01276) with Roger Howe (Yale/Texas A&M). We introduce a language to speak about "size" of a representation, and we develop a method (called "eta correspondence") that produces analytic information on (conjecturally all the) "small" representations of finite classical groups.

The talk should be accessible to anyone with basic linear algebra knowledge. I will illustrate our theory with concrete motivations and numerical data obtained with John Cannon (MAGMA, Sydney) and Steve Goldstein (Scientific computing, Madison).