Let F be a nonarchimedean local field, and G a connected reductive group defined over F . We classify the representations of G(F ) that contain any anisotropic unrefined minimal K-type satisfying a certain tameness condition. We show that these representations are induced from compact (mod center) subgroups, and we construct corresponding refined minimal K-types.

ABSTRACT. We show that, under fairly general conditions, many elements of a p-adic group can be well approximated by a product whose factors have properties that are helpful in performing explicit character computations.

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Abstract. 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. IntroductionLet 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 Θ π of π is the map C ∞ c (G) → C given by Θ π (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 Θ π is represented by a locally constant function on the set of regular semisimple elements in G. We will denote this function also by Θ π .For any semisimple γ ∈ G, the local character expansion about γ (see [11] and [10]) is the identityvalid for all regular semisimple Y in the Lie algebra m of the centralizer of γ such that Y is close enough to 0. Here, the sum is over the set of nilpotent orbits O in m; µ O is the function that represents the distribution that is the Fourier transform of the orbital integral µ O associated to O; c O = c O,γ (π) ∈ C; and e is the exponential map, or some suitable substitute. This is a qualitative result, in the sense that it gives no indication of how close Y must be to 0 in order for the identity to be valid. Many questions in harmonic analysis on G require more quantitative versions of such qualitative results. As an example of a quantitative result, DeBacker [6] has determined (under some hypotheses on G) a neighborhood of validity for the local character expansion near the identity, thus verifying a conjecture of Hales, Moy, and Prasad (see [15]).1991 Mathematics Subject Classification. 22E35, 22E50, 20G25.

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