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.     

On the basic algebraic operations in solvable Lie groups

Summary For a simply connected solvable Lie group we specify the structure of the product, the inverse and the exponential map expressed in suitable coordinates (canonical coordinates of the second kind), and point out that in these coordinates the product and inverse are expressed entirely in terms of polynomials, exponential functions and trigonometric functions. We devise algorithms for computing the product, the inverse and the exponential map.     

Moment sets and unitary dual for the diamond group

The diamond group G is a solvable group, semi-direct product of R with a (2n+1)-dimensional Heisenberg group Hn. We consider this group as a first example of a semi-direct product with the form R?N where N is nilpotent, connected and simply connected.Computing the moment sets for G, we prove that they separate the coadjoint orbits and its generic unitary irreducible representations.Then we look for the separation of all irreducible representations. First, moment sets separate representations for a quotient group G⁻ of G by a discrete subgroup, then we can extend G to an overgroup G⁺, extend simultaneously each unitary irreducible representation of G to G⁺ and separate the representations of G by moment sets for G⁺.     

Stability of the local gamma factor in the unitary case

In [Stephen Rallis, David Soudry, Stability of the local gamma factor arising from the doubling method, Math. Ann. 333 (2) (2005) 291-313, MR2195117 (2006m:22026)], Rallis and Soudry prove the stability under twists by highly ramified characters of the local gamma factor arising from the doubling method, in the case of a symplectic group or orthogonal group G over a local non-archimedean field F of characteristic zero, and a representation π of G, which is not necessarily generic. This paper extends their arguments to show the stability in the case when G is a unitary group over a quadratic extension E of F, thereby completing the proof of the stability for classical groups. This stability property is important in Cogdell, Kim, Piatetski-Shapiro, and Shahidi's use of the converse theorem to prove the existence of a weak lift from automorphic, cuspidal, generic representations of G(A) to automorphic representations of GLn(A) for appropriate n, to which references are given in [Stephen Rallis, David Soudry, Stability of the local gamma factor arising from the doubling method, Math. Ann. 333 (2) (2005) 291-313, MR2195117 (2006m:22026)].     

A note on rationality of orbital integrals on a P-adic group

We prove that if rational measures are used on p-adic reductive groups then the orbital integrals of any given smooth and compactly supported complex valued function belong to the field generated by the values of that function. We also show that the Shalika germs are then rational valued functions. As a consequence, we are able to show, in certain cases, that the coefficients appearing in the Harish-Chandra local character expansion are rational numbers. Research supported by NSERC     

Dualité dans l’espace des caractères virtuels tempérés d’un groupe réductif <Emphasis Type="Italic">p</Emphasis>-adique

Résumé We define an involution on the set of tempered virtual characters of a connected reductive p-adic group. This involution commutes with character of parabolic induction and with truncation. It also preserves the irreducible characters up to sign and the elliptic inner product.     

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.     

On minimal representations of Chevalley groups of type D n , E n and G 2

Let G be a simply connected Chevalley group of type D _n , E _n or G₂. In this paper, we show that the minimal representation of G is unique for types D _n and E _n and it does not exist for the type G₂.     

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.     

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.     

On the properly discontinuous subgroups of affine motions

The fundamental groupΓ of a compact complete affine manifold is represented as an affine crystallographic subgroup of Aff(n). L.S.Auslander conjectured thatΓ is virtually solvable. Our purpose is to find the algebraic condition onΓ which leads affirmative answer to the conjecture.     

Reproducing kernels for harmonic Bergman spaces of the unit ball

We study reproducing kernels for harmonic Bergman spaces of the unit ball inR ⁿ . We establish some new properties for the reproducing kernels and give some applications of these properties.     

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.     

Branching Rules, Kostka–Foulkes Polynomials and q-multiplicities in Tensor Product for the Root Systems B n , C n and D n

The Kostka–Foulkes polynomials related to a root system can be defined as alternating sums running over the Weyl group associated to . By restricting these sums over the elements of the symmetric group when is of type or , we obtain again a class of Kostka–Foulkes polynomials. When is of type or there exists a duality between these polynomials and some natural -multiplicities and in tensor products [11]. In this paper we first establish identities for the which implies in particular that they can be decomposed as sums of Kostka–Foulkes polynomials with nonnegative integer coefficients. Moreover these coefficients are branching coefficients This allows us to clarify the connection between the -multiplicities and the polynomials defined by Shimozono and Zabrocki. Finally we show that and coincide up to a power of with the one dimension sum introduced by Hatayama and co-workers when all the parts of are equal to , which partially proves some conjectures of Lecouvey and Shimozono and Zabrocki.Presented by P. Littelmann.     

Unitary highest weight representations via the orbit method I: The scalar case

We use Kirillov's orbit method to construct all unitarizable highest weight representations with scalar lowest K-type.     

On the structure of Bergman and poly-Bergman spaces

Let II be the upper half-plane in, consider the Bergman space , the subspace of all analytic functions fromL ₂(II). The complete decomposition ofL ₂(II) onto Bergman and Bergman type spaces of poly-analytic and poly-anti-analytic functions is obtained. The orthogonal Bergman type projections onto each of these subspaces are described. Connections with the Hardy spaces and the Szegö projections are established.This work was partially supported by CONACYT Project 3114P-E9607, México.