Green Functions and Higher 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 prove some basic properties along the lines of the finite field case, like a summation formula. As an application we show that the higher Deligne-Lusztig characters and Gérardin's characters agree at regular semisimple elements.