The Local Character Expansion near a Tame, Semisimple Element

Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses, we describe an explicit region on which the local character expansion is valid. We assume neither that the group is connected, nor that the underlying field has characteristic zero. 0. Introduction Let G denote the group of k-points of a reductive k-group G, where k is a nonarchimedean local field. To simplify the present discussion, assume for now that G is connected and that k has characteristic zero. Let (π, V ) denote an irreducible admissible representation of G. Let dg denote a fixed Haar measure on G, and let C c (G) denote the space of complex-valued, locally constant, compactly supported functions on G. The distribution character Θπ of π is the map C ∞ c (G) → C given by Θπ(f) := tr π(f), where π(f) is the (finite-rank) operator on V given by π(f)v := ∫ G f(g)π(g)v dg. From Howe [12] and Harish-Chandra [9], the distribution Θπ is represented by a locally constant function on the set of regular semisimple elements in G. We will denote this function also by Θπ. For any semisimple γ ∈ G, the local character expansion about γ (see [11] and [10]) is the identity Θπ(γ e(Y )) = ∑