Harmonic Analysis on GL(n) over Finite Fields

Understand

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(\pi(g)) / dim(\pi), for an irreducible representation \pi of G and an element g of G.
• It turns out [Gurevich-Howe15, Gurevich-Howe17] that for classical groups G over finite fields there are several (compatible) invariants of representations that provide strong information on the character ratios.
• We call these invariants collectively rank.