Allen Moy

Unrefined minimal K–types for p–adic groups

via their restriction to compact open subgroups was begun by Mautner, Shalika and Tanaka for groups of type AI. In contrast to real reductive groups where the representation theory of a maximal compact subgroup K is given in terms of slight modifications to Cartans theory of the highest weight, the representation theory of a (compact) parahoric subgroup ,~ is quite complicated. There is still no comprehensive theory for classifying the irreducible representa- tions. In the case of

Jacquet functors and unrefined minimal K-types

The notion of an unrefined minimal K-type is extended to an arbitrary reductive group over a non archimedean local field. This allows one to define the depth of a representation. The relationship between unrefined minimal K-types and the functors of Jacquet is determined. Analogues of fundamental results of Borel are proved for representations of depth zero.

Reduction to real infinitesimal character in affine Hecke algebras

The main result of this paper is to show that the problem of the determination of the unramified unitary dual of a split p-adic group is equivalent to the problem of determining the unitary dual of the corresponding graded Hecke algebra. In [BM], the authors established this equivalence in the case of Iwahori spherical representations under a certain restriction; namely, it was essential for the infinitesimal character to be real (in the terminology of [BMJ). In terms of the Langlands-Deligne-Lusztig parameters (s, u, p) [KUL, the restriction is that S E LG be a purely hyperbolic element. The technique used in [BM] was to combine the notion of the signature of a K-character in [V] with some facts which follow from [KL], namely, that the 2w-characters of tempered representations are linearly independent. This is essentially true precisely when the infinitesimal character is real; for if not, s has an elliptic part Se such that

Unitary spherical spectrum for p-adic classical groups

Let G be a split reductive p-adic group. Then the determination of the unitary representations with nontrivial Iwahori fixed vectors can be reduced to the determination of the unitary dual of the corresponding Iwahori-Hecke algebra. In this paper we study the unitary dual of the Iwahori-Hecke algebras corresponding to the classical groups. We determine all the unitary spherical representations.

Local character expansions

Abstract In this paper the authors develop a method to compute the local character expansion of a depth zero representation of a p-adic group. The main idea is to use the generalized Gelfand-Graev characters for finite groups as test functions to plug into the character formula. This is possible due to results of Waldspurger on the validity of the local character expansion in a large enough neighborhood of the identity. The method leads to a classification of the unipotent orbits in terms of parahoric subalgebras.

The Bernstein center in terms of invariant locally integrable functions

1.1. The Lie algebra g and associated enveloping algebra U(g)of a connected Lie group G are fundamental tools in the study of the group’s representations. The center Z(U(g)) of the enveloping algebra is particularly useful for a number of purposes. There are three useful descriptions of the center: (i)an algebraic description of the center in terms of the Lie algebra, (ii)the Harish-Chandra homomorphism identification of the center with the space of Weyl invariant regular functions on the symmetric algebra of a Cartan subalgebra, and (iii)as invariant distributions on the group supported at the identity. 1.2. One can also, following J. Bernstein, consider the center Z(U(g)) from a ring and categorical point of view. If A is an algebra with identity, then the center of A is naturally isomorphic to the algebra of endomorphisms of the identity functor of the category of all A-modules. In particular, Z(U(g)) is isomorphic to the algebra of endomorphisms of the identity functor category of modules of the Lie algebra. This formulation of the center of the enveloping algebra is the starting point for Bernstein’s construction of an analogue of the center of the enveloping algebra for reductive p-adic groups. If G is a reductive p-adic group G, let S(G)denote the category of smooth representations of G. The Bernstein center Z(G)of G is defined to be the algebra of endomorphisms of the identity functor of S(G)There are two realizations of Bernstein center. Both are all much less concrete than in the real case. (i)The most explicit description of the Bernstein center of G is in terms of algebraic

A new proof of the Howe Conjecture

Let J(Q) denote the space of g-invariant distributions supported on Ad(g9)(Q). If g E g, set Ad(g)f: z F-> f(g`zg). If q is a locally constant compactly supported function on g (resp. g), define TQ(q) to be the linear functional on J(Q) defined by TQ(O)(T) := T(0). The set Q will be fixed throughout the paper, so we will abbreviate TQ to T. The g-invariance of the distributions in J(Q) means T(Ad(g)f) = T(f). If L is a compact open subgroup of g, denote as C,(g/L) the space of compactly supported functions on g which are right L-invariant. Similary, if L is an open compact lattice in g, let C,(g/L) be the space of compactly supported functions on g which are L-invariant. In [Hol], Howe conjectured that if Q is a compact subset of g (resp. 9), then the restriction of the distributions in J(Q) to C,(B/L) (resp. C,(g/L)) is finite dimensional. A more common formulation of the group version of the Howe conjecture replaces the space C,(g/L) with the Hecke algebra 7-(g//L) of compactly supported L-biinvariant functions on g. Since 7-(g//L) c C,(g/L), apriori, our formulation with C,(g/L) implies the Hecke algebra formulation. It is a simple matter to show that in fact the two formulations are equivalent. If f E C,(9/L), then the function F(g) := Lf (kgk-1)dk belongs to the Hecke algebra, and the map f 1-* F is a surjection. The linear functional on J(Q) induced by f is a positive multiple of the linear functional induced by F. Thus, the two spaces of linear functionals are equal and so the two formulations are equivalent.

Unipotent representations and reductive dual pairs over finite fields

Consider the representation correspondence for a reductive dual pair (G 1 , G 2 ) over a finite field. We consider the question of how the correspondence behaves for unipotent representations. In the special case of cuspidal unipotent representations, and a certain fundamental situation, that of «first occurrence», the representation correspondence takes a cuspidal unipotent representation of G 1 to one of G 2 . This should serve as a fundamental case in studying the correspondence in general over both finite and local fields

The irreducible orthogonal and symplectic Galois representations of a p-adic field (The tame case)

Abstract Let F be a p-adic field. If n is a natural number relatively prime to p, then all the irreducible n-dimensional Galois representations are parametrized by admissible characters. This parametrization is used to determine which of these characters are real-valued, and among the real-valued representations to distinguish the orthogonal representations from the symplectic representations.

Dirac cohomology of one-W -type representations

The smooth hermitian representations of a split reductive p-adic group whose restriction to a maximal hyperspecial compact subgroup contain a single K-type with Iwahori fixed vectors have been studied in [D. Barbasch, A. Moy, Classification of one K-type representations, Trans. Amer. Math. Soc. 351 (1999), no. 10, 4245-4261] in the more general setting of modules for graded affine Hecke algebras with parameters. We show that every such one K-type module has nonzero Dirac cohomology (in the sense of [D. Barbasch, D. Ciubotaru, P. Trapa, The Dirac operator for graded affine Hecke algebras, arXiv:1006.3822]), and use Dirac operator techniques to determine the semisimple part of the Langlands parameter for these modules, thus completing their classification.