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Dan Barbasch

Publications and source records attributed to Dan Barbasch.

31 records · Page 2Linked to original sources
arXiv2015

Hermitian forms for affine Hecke algebras

We study star operations for Iwahori-Hecke algebras and invariant hermitian forms for finite dimensional modules over (graded) affine Hecke algebras with a view towards a unitarity algorithm.

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arXiv2014

Ladder representations of GL(n,Q_p)

In this paper, we recover certain known results about the ladder representations of GL(n, Q_p) defined and studied by Lapid, Minguez, and Tadic. We work in the equivalent setting of graded Hecke algebra modules. Using the Arakawa-Suzuki functor from category O to graded Hecke algebra modules, we show that the determinantal formula proved by Lapid-Minguez and Tadic is a direct consequence of the BGG resolution of finite dimensional simple gl(n)-modules. We make a connection between the semisimplicity of Hecke algebra modules, unitarity with respect to a certain hermitian form, and ladder representations.

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arXiv2012

Generic unipotent standard modules

Using Lusztig's geometric classification, we find the reducibility points of a standard module for the affine Hecke algebra, in the case when the inducing data is generic. This recovers the known result of Muic-Shahidi for representations of split p-adic groups with Iwahori-spherical Whittaker vectors. We also give a necessary (insufficient) condition for reducibility in the non-generic case.

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arXiv2011

Unitary equivalences for reductive p-adic groups

We establish a transfer of unitarity for a Bernstein component of the category of smooth representations of a reductive p-adic group to the associated Hecke algebra, in the framework of the theory of types, whenever the Hecke algebra is an affine Hecke algebra with geometric parameters, in the sense of Lusztig (possibly extended by a group of automorphisms of the root datum). It is known that there is a large class of such examples (detailed in the paper). As a consequence, we establish relations between the unitary duals of different groups, in the spirit of endoscopy.

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arXiv2010

Dirac cohomology for graded affine Hecke algebras

We define analogues of the Casimir and Dirac operators for graded affine Hecke algebras, and establish a version of Parthasarathy's Dirac operator inequality. We then prove a version of Vogan's Conjecture for Dirac cohomology. The formulation of the conjecture depends on a uniform geometric parametrization of spin representations of Weyl groups. Finally, we apply our results to the study of unitary representations.

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arXiv2009

Unitary spherical spectrum for split classical groups

This paper gives a complete classification of the unitary irreducible spherical representations of split real and p-adic groups. The results were obtained around 2000, the changes to the new version are expository.

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arXiv2008

Cuspidal representations of reductive groups

This paper proves the existence of cuspidal automorphic forms for a reductive group, invariant under an automorphism of finite order. The techniques used are a local analysis of orbital integrals and the Arthur-Selberg trace formula.

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arXiv2008

Whittaker unitary dual of affine graded Hecke algebras of type E

This paper gives the classification of the Whittaker unitary dual for affine graded Hecke algebras of type E. By the Iwahori-Matsumoto involution, this is equivalent also to the classification of the spherical unitary dual for type E. This work completes the classification of the Whittaker Iwahori-spherical unitary dual, or equivalently, the spherical unitary dual, of split linear algebraic p-adic groups.

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arXiv2008

Unitarizable minimal principal series of reductive groups

The aim of this paper is to give an exposition of recent progress on the determination of the unitarizable Langlands quotients of minimal principal series for reductive groups over the real or p-adic fields in characteristic 0.

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