Traces on Iwahori-Hecke algebras and counting rational points

TL;DR

This paper links the count of rational points on certain varieties associated with reductive groups over finite fields to traces on Iwahori-Hecke algebras, revealing geometric properties when elements are elliptic.

Contribution

It establishes a connection between rational point counts and algebraic traces, and characterizes the geometric structure of these varieties for elliptic elements.

Findings

Number of rational points expressed via Iwahori-Hecke algebra trace

Variety is smooth and irreducible for elliptic, minimal length elements

Provides geometric insights into varieties linked with Weyl group elements

Abstract

Let w be an element of the Weyl group of a reductive group G defined and split over a finite field. We consider the variety of triples (g,B,B') where g is a unipotent element of G and B, B' are Borel subgroups of G such that B contains g and B',gB'g^{-1} are in relative position w. We show that the number of rational points of this variety can be expressed in terms of a trace on the Iwahori-Hecke algebra. We also show that this variety is smooth, irreducible, if w is elliptic, of minimal length in its conjugacy class.