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.     

Construction of tame supercuspidal representations

We give a quite general construction of irreducible supercuspidal representations and supercuspidal types (in the sense of Bushnell and Kutzko) of -adic groups. In the tame case, the construction should include all known constructions, and it is expected that this gives all supercuspidal representations. We also give a conjectural Hecke algebra isomorphism, which can be used to analyze arbitrary irreducible admissible representations, following the ideas of Howe and Moy.

     

Gamma factors of certain supercuspidal representations

A -factor defined by the doubling method is calculated for certain supercuspidal representations. The result is a sum over a finite group of Lie type, which may be called a non-abelian Gauss sum. It is the sum of the product of a cuspidal character and an Igusa zeta integral. In some cases, we show that our result agrees with what is expected by the local Langlands conjecture. Received: 11 July 1999 / Accepted: 29 December 1999 / Published online: 28 June 2000     

Symplectic supercuspidal representations and related problems

In this paper, we summarize the basic structures and properties of irreducible symplectic supercuspidal representations of GLn(F) over a p-adic local field F with characteristic zero, and explore possible topics for further investigation.     

The characters of supercuspidal representations as weighted orbital integrals

Weighted orbital integrals are the terms which occur on the geometric side of the trace formula. We shall investigate these distributions on ap-adic group. We shall evaluate the weighted orbital integral of a supercuspidal matrix coefficient as a multiple of the corresponding character. Supported in part by NSERC Operating Grant A3483.     

Self-contragredient supercuspidal representations of

Let be a non-archimedean local field of residual characteristic . Then has tamely ramified self-contragredient supercuspidal representations if and only if or is even. When such representations exist, they do so in abundance.

     

The multiplicity one conjecture for local theta correspondences

Over a non-archimedean local field of characteristic zero, we prove multiplicity preservation of local theta correspondences for orthogonal-symplectic dual pairs. The same proof works for dual pairs of unitary groups.     

Murnaghan-Kirillov theory for depth-zero supercuspidal representations: reduction to Lusztig functions

The goal of Murnaghan-Kirillov theory is to associate to an irreducible smooth representation of a reductive p-adic group a family of regular semisimple orbital integrals in the Lie algebra with the following property: the character of ?? is given, on a well determined set, by an explicit combination of the Fourier transforms of these orbital integrals. Subject to certain restrictions, we adapt arguments of Waldspurger to show that, for depth-zero irreducible smooth supercuspidal representations, this problem may be reduced to a similar one for distributions associated to Lusztig functions.     

Branching rules obtained from explicit correspondences of automorphic representations

In this paper, we show how one can use the theory of lifting of automorphic forms in order to obtain formulas of branching rules in complex groups. We also use this technique to give formulas for certain functionals which are defined on unramified representations.     

Local group representations

As a generalization of the usual notion of a group representation, a local group representation of connected and locally connected groups G is introduced as a certain sheaf A with base G, such that all fibres A _x may be identified with a fixed linear subspace D of the representation space. The sheaf provides a means of expressing a group symmetry which is known to hold only locally (in a sense which will be made precise). The problem of extending a local representation to a global one may be considered as a noncommutative counterpart of the dilation theory of systems of doubly commuting contractions. If G is a Lie group, one may associate a derived representation of the enveloping algebra (g) of the Lie algebra g of G. In case of isometric local representations in a Hilbert space, these are *-representations and determine the appropriate local representation uniquely. As an application we obtain that the n-tuples of antisymmetric operators of partial derivatives on L ²(), where is a connected open subset of ⁿ, determine up to translation.     

Exterior square gamma factors for cuspidal representations of $$\mathrm {GL}_n$$ GL n : simple supercuspidal representations

We compute the local twisted exterior square gamma factors for simple supercuspidal representations, using which we prove a local converse theorem for simple supercuspidal representations.

     

Liouville correspondences between multicomponent integrable hierarchies

Theoretical and Mathematical Physics - We establish Liouville correspondences for the integrable two-component Camassa–Holm hierarchy, the two-component Novikov $$($$ Geng–Xue $$)$$...     

Explicit functorial correspondences for level zero representations of p-adic linear groups

Let F be a non-Archimedean local field and D a central F-division algebra of dimension n², n?1. We consider first the irreducible smooth representations of D^× trivial on 1-units, and second the indecomposable, n-dimensional, semisimple, Weil-Deligne representations of F which are trivial on wild inertia. The sets of equivalence classes of these two sorts of representations are in canonical (functorial) bijection via the composition of the Jacquet-Langlands correspondence and the Langlands correspondence. They are also in canonical bijection via explicit parametrizations in terms of tame admissible pairs. This paper gives the relation between these two bijections. It is based on analysis of the discrete series of the general linear group GLn(F) in terms of a classification by extended simple types.     

Unitary representations and theta correspondence for type I classical groups

In this paper, we discuss the positivity of the Hermitian form (,)_π introduced by Li in Invent. Math. 27 (1989) 237-255. Let (G₁,G₂) be a type I dual pair with G₁ the smaller group. Let π be an irreducible unitary representation in the semistable range of θ(MG₁,MG₂) (see Communications in Contemporary Mathematics, Vol. 2, 2000, pp. 255-283). We prove that the invariant Hermitian form (,)_π is positive semidefinite under certain restrictions on the size of G₂ and a mild growth condition on the matrix coefficients of π. Therefore, if (,)_π does not vanish, θ(MG₁,MG₂)(π) is unitary.Theta correspondence over was established by Howe in (J. Amer. Math. Soc. 2 (1989) 535-552). Li showed that theta correspondence preserves unitarity for dual pairs in stable range. Our results generalize the results of Li for type I classical groups (Invent. Math. 27 (1989) 237). The main result in this paper can be used to construct irreducible unitary representations of classical groups of type I.