Colloquium abstracts

Shamgar Gurevich
University of Wisconsin, USA
October 1, 2020

Harmonic Analysis on GLn 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 Trace$( ho(g)) / dim( ho), for an irreducible representation ho of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G. Recently, we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank. Rank suggests a new organization of representations based on the very few `small' ones. This stands in contrast to Harish-Chandra's `philosophy of cusp forms', which is (since the 60s) the main organization principle, and is based on the (huge collection of) `Large' representations. This talk will discuss the notion of rank for the group GLn over finite