On the Steinberg character of a semisimple p-adic group

Abstract

We show that the character of the Steinberg representation of a split semisimple p-adic group at a very regular element is given (up to sign) by a power of q, the number of elements in the residue field. We also show that (under an assumption on the characteristic) the character of an Iwahori-spherical representation at a split very regular element is given by a trace in the corresponding Hecke algebra module.