  Let G be a semisimple almost simple algebraic group defined and split over a
nonarchimedean local field K and let S be the Steinberg representation of G(K).
Let t be be a very regular semisimple element of G(K). In this paper we give a
simple formula (not as an alternating sum) for the value of the character of S
at t; we show that this value is plus or minus an integer power of q, the
cardinal of the residue field of K. We also give an explicit formula for the
character at a split very regular element of any irreducible admissible
representation of G(K) with nonzero vectors invariant under an Iwahori subgroup
(valid under some restrictions on characteristic).

Representation Theory