Kazhdan-Laumon representations of finite Chevalley groups, character sheaves and some generalization of the Lefschetz-Verdeier trace formula

TL;DR

This paper explores Kazhdan-Laumon representations of finite Chevalley groups, showing their generic irreducibility and connection to Lusztig's character sheaves, and provides a geometric construction of their isomorphism under a generalized trace formula.

Contribution

It demonstrates the irreducibility of Kazhdan-Laumon representations and links them to Lusztig's character sheaves, also offering a geometric construction of their isomorphism.

Findings

Generic Kazhdan-Laumon representations are irreducible.

Their characters match traces of Frobenius on Lusztig's sheaves.

They are isomorphic to Deligne-Lusztig representations.

Abstract

In this paper we investigate the representations of reductive groups over a finite field, introduced in 1987 by D.Kazhdan and G.Laumon. We show that generically these representations are irreducible and that their character is equal to the function obtained by traces of Frobenius morphism on the stalks of the corresponding Lusztig's character sheaf. This, together with the corresponding result by G.Lusztig, implies that generic Kazhdan-Laumon representations are isomorphic to the corresponding Deligne-Lusztig representations. In the second half of the paper we give a direct geometric construction of this isomorphism, assuming that certain generalization of the Lefschetz-Verdier trace formula holds.