ON THE STEINBERG CHARACTER OF A SEMISIMPLE p-ADIC GROUP Ju-Lee Kim and George Lusztig

1.1. Let K be a nonarchimedean local field and let K be a maximal unramified field extension of K. Let O (resp. O) be the ring of integers of K (resp. K) and let p (resp. p) be the maximal ideal of O (resp. O). Let K = K −{0}. We write O/p = Fq, a finite field with q elements, of characteristic p. Let G be a semisimple almost simple algebraic group defined and split over K with a given O-structure compatible with the K-structure. If V is an admissible representation of G(K) of finite length, we denote by φV the character of V in the sense of Harish-Chandra, viewed as a C-valued function on the set G(K)rs := Grs ∩ G(K). (Here Grs is the set of regular semisimple elements of G and C is the field of complex numbers.) In this paper we study the restriction of the function φV to: (a) a certain subset G(K)vr of G(K)rs, that is to the set of very regular elements in G(K) (see 1.2), in the case where V is the Steinberg representation of G(K) and (b) a certain subset G(K)svr of G(K)vr, that is to the set of split very regular elements in G(K) (see 1.2), in the case where V is an irreducible admissible representation of G(K) with nonzero vectors fixed by an Iwahori subgroup. In case (a) we show that φV (g) with g ∈ G(K)rs is of the form ±qn with n ∈ {0,−1,−2, . . .} (see Corollary 3.4) with more precise information when g ∈ G(K)svr (see Theorem 2.2) or when g ∈ G(K)cvr (see Theorem 3.2); in case (b) we show (with some restriction on characteristic) that φV (g) with G(K)svr can be expressed as a trace of a certain element of an affine Hecke algebra in an irreducible module (see Theorem 4.3). Note that the Steinberg representation S is an irreducible admissible representation of G(K) with a one dimensional subspace invariant under an Iwahori