Semisimple strata for p-adic classical groups

Let F₀ be a non-archimedean local field, of residual characteristic different from 2, and let G be a unitary, symplectic or orthogonal group defined over F₀. In this paper, we prove some fundamental results towards the classification of the representations of G via types [8]. In particular, we show that any positive level supercuspidal representation of G contains a semisimple skew stratum, that is, a special character of a certain compact open subgroup of G. The intertwining of such a stratum has been calculated in [19].     

Asymptotics and integrability properties for matrix coefficients of admissible representations of reductive p-adic groups

We obtain the necessary and sufficient conditions for the pth power integrability of a matrix coefficient of an admissible representation of a reductive p-adic group in terms of the exponents of the matrix coefficient.     

A geometric construction of intertwining operators for reductive p-adic groups

In this paper we construct standard intertwining operators for reductive p-adic groups by a method of Bernstein.     

Polynomial representations of the orthogonal groups

This paper is concerned with realizations of the irreducible representations of the orthogonal group and construction of specific bases for the representation spaces. As is well known, Weyl's branching theorem for the orthogonal group provides a labeling for such bases, called Gelfand-Žetlin labels. However, it is a difficult problem to realize these representations in a way that gives explicit orthogonal bases indexed by these Gelfand-–etlin labels. Thus, in this paper the irreducible representations of the orthogonal group are realized in spaces of polynomial functions over the general linear groups and equipped with an invariant differentiation inner product, and the Gelfand-Žetlin bases in these spaces are constructed explicitly. The algorithm for computing these polynomial bases is illustrated by a number of examples. Partially supported by a grant from the Department of Energy. Partially supported by NSF grant No. MCS81-02345.     

Polynomial representations of the symplectic groups

The irreducible finite dimensional representations of the symplectic groups are realized as polynomials on the irreducible representation spaces of the corresponding general linear groups. It is shown that the number of times an irreducible representation of a maximal symplectic subgroup occurs in a given representation of a symplectic group, is related to the betweenness conditions of representations of the corresponding general linear groups. Using this relation, it is shown how to construct polynomial bases for the irreducible representation spaces of the symplectic groups in which the basis labels come from the representations of the symplectic subgroup chain, and the multiplicity labels come from representations of the odd dimensional general linear groups, as well as from subgroups. The irreducible representations of Sp(4) are worked out completely, and several examples from Sp(6) are given.     

Square integrability of representations on p-adic symmetric spaces

A symmetric space analogue of Casselman's criterion for square integrability of representations of a p-adic group is established. It is described in terms of exponents of Jacquet modules along parabolic subgroups associated to the symmetric space.     

Isolatedness of the minimal representation and minimal decay of exceptional groups

Using a new definition of rank for representations of semisimple groups sharp results are proved for the decay of matrix coefficients of unitary representations of two types of non-split p-adic simple algebraic groups of exceptional type. These sharp bounds are achieved by minimal representations. It is also shown that in one of the cases considered, the minimal representation is isolated in the unitary dual.     

Classical and quantum Heisenberg groups, their representations and applications

This paper is a survey on classical Heisenberg groups and algebras, q-deformed Heisenberg algebras, q-oscillator algebras, their representations and applications. Describing them, we tried, for the reader's convenience, to explain where the q-deformed case is close to the classical one, and where there are principal differences. Different realizations of classical Heisenberg groups, their geometrical aspects, and their representations are given. Moreover, relations of Heisenberg groups to other linear groups are described. Intertwining operators for different (Schrödinger, Fock, compact) realizations of unitary irreducible representations of Heisenberg groups are given in explicit form. Classification of irreducible representations and representations of the q-oscillator algebra is derived for the cases when q is not a root of unity and when q is a root of unity. The Fock representation of the q-oscillator algebra is studied in detail. In particular, q-coherent states are described. Spectral properties of some operators of the Fock representations of q-oscillator algebras are given. Some of applications of Heisenberg groups and algebras, q-Heisenberg algebras and q-oscillator algebras are briefly described.     

Compositions of theta correspondences

Theta correspondence θ over is established by Howe (J. Amer. Math. Soc. 2 (1989) 535). In He (J. Funct. Anal. 199 (2003) 92), we prove that θ preserves unitarity under certain restrictions, generalizing the result of Li (Invent. Math. 97 (1989) 237). The goal of this paper is to elucidate the idea of constructing unitary representation through the propagation of theta correspondences. We show that under a natural condition on the sizes of the related dual pairs which can be predicted by the orbit method (J. Algebra 190 (1997) 518; Representation Theory of Lie Groups, Park City, 1998, pp. 179-238; The Orbit Correspondence for real and complex reductive dual pairs, preprint, 2001), one can compose theta correspondences to obtain unitary representations. We call this process quantum induction.     

Torsion p-adic Galois representations and a conjecture of Fontaine

Let p be a prime, K a finite extension of Qp and T a finite free Zp-representation of . We prove that TZp_⊗Qp is semi-stable (resp. crystalline) with Hodge-Tate weights in {0,…,r} if and only if, for all n, T/pnT is torsion semi-stable (resp. crystalline) with Hodge-Tate weights in {0,…,r}.     

Lowest weight representations of some infinite dimensional groups on fock spaces

The results of Kashiwara and Vergne on the decomposition of the tensor products of the Segal-Shale-Weil representation are extended to the infinite dimensional case and give all unitary lowest weight representations. Our methods are basically algebraic. When restricted to the finite dimensional case, they yield a new proof.     

Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties

Let G be a reductive group defined over a p-adic local field L, let P be a parabolic subgroup of G with Levi quotient M, and write G:=G(L), P:=P(L), and M:=M(L). In this paper we construct a functor JP from the category of essentially admissible locally analytic G-representations to the category of essentially admissible locally analytic M-representations, which we call the Jacquet module functor attached to P, and which coincides with the usual Jacquet module functor of [Casselman W., Introduction to the theory of admissible representations of p-adic reductive groups, unpublished notes distributed by P. Sally, draft dated May 7, 1993. Available electronically at http://www.math.ubc.ca/people/faculty/cass/research.html. [5]] on the subcategory of admissible smooth G-representations. We establish several important properties of this functor.