This paper studies the behavior of Jiu-Kang Yu's tame supercuspidal representations relative to involutions of reductive p-adic groups. Symmetric space methods are used to illuminate various aspects of Yu's construction. Necessary conditions for a tame supercuspidal representation of G to be distinguished by (the fixed points of) an involution of G are expressed in terms of properties of the G-orbit of the associated G-datum. When these conditions are satisfied, the question of whether a tame supercuspidal representation is distinguished reduces to the question of whether certain cuspidal representations of finite groups of Lie type are distinguished relative to particular quadratic characters. As an application of the main results, we obtain necessary and sufficient conditions for equivalence of two of Yu's supercuspidal representations associated to distinct G-data.

Distinguished tame supercuspidal representations

Let G be a hermitian quaternionic group. We determine complementary series for representations of G induced from super-cuspidal representations of a Levi factor of the Siegel maximal parabolic subgroup of G.

Complementary Series for Hermitian Quaternionic Groups

This paper analyzes the space HomH(π, 1), where π is an irreducible, tame supercuspidal representation of GL(n) over a p-adic field and H is a unitary group in n variables contained in GL(n). It is shown that this space of linear forms has dimension at most one. The representations π which admit nonzero H-invariant linear forms are characterized in several ways, for example, as the irreducible, tame supercuspidal representations which are quadratic base change lifts.

/pdf/tame-supercuspidal-representations-of-gl-n-distinguished-by-uawf0inq6x.pdf

Tame Supercuspidal Representations of GL( n ) Distinguished by a Unitary Group

In this paper we determine the structure of the full lift for the Howe correspondence of for rank-one reducibilities.

On the Structure of the Full Lift for the Howe Correspondence of $(Sp(n), O(V))$ for Rank-One Reducibilities

In this paper we study reducibility of representations of split classical p-adic groups induced from self-contragredient supercuspidal represetation associated via Howe's construction to an admissible character, we show that in many cases Shahidi's criterion for reducibility of the induced representation reduces to a simple condition on the admissible character.

/pdf/reducibility-of-some-induced-representations-of-split-3x8a0xxx3z.pdf

Reducibility of some Induced Representations of Split Classical p-Adic Groups

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Reducibility of some Induced Representations of Split Classical p-Adic Groups