Green functions and higher Deligne--Lusztig characters

TL;DR

This paper extends Deligne--Lusztig character formulas to local rings, introduces Green functions in this context, and demonstrates their properties and applications, including character comparisons and gamma function generalizations.

Contribution

It generalizes Deligne--Lusztig character formulas to truncated formal power series rings and defines Green functions for these local rings.

Findings

Higher Deligne--Lusztig and G"erardin's characters agree at regular semisimple elements

Established summation formula for Green functions in local rings

Generalized gamma function results for Deligne--Lusztig characters

Abstract

We give a generalisation of the character formula of Deligne--Lusztig representations from the finite field case to the truncated formal power series case. Motivated by this generalisation, we give a definition of Green functions for these local rings, and we prove some basic properties along the lines of the finite field case, like a summation formula. Among the applications we show that the higher Deligne--Lusztig characters and G\'erardin's characters agree at regular semisimple elements. We also derive a generalisation of Braverman and Kazhdan's result on gamma functions for Deligne--Lusztig characters, with a more elementary argument.