Hecke algebras with unequal parameters and Vogan's left cell invariants

In 1979, Vogan introduced a generalised $\\tau$ -invariant for characterising primitive ideals in enveloping algebras. Via a known dictionary this translates to an invariant of left cells in the sense of Kazhdan and Lusztig. Although it is not a complete invariant, it is extremely useful in describing left cells. Here, we propose a general framework for defining such invariants which also applies to Hecke algebras with unequal parameters.Comment: 15 pages. arXiv admin note: substantial text overlap with arXiv:1405.573

Dirac cohomology, elliptic representations and endoscopy

The first part (Sections 1-6) of this paper is a survey of some of the recent developments in the theory of Dirac cohomology, especially the relationship of Dirac cohomology with (g,K)-cohomology and nilpotent Lie algebra cohomology; the second part (Sections 7-12) is devoted to understanding the unitary elliptic representations and endoscopic transfer by using the techniques in Dirac cohomology. A few problems and conjectures are proposed for further investigations.Comment: This paper will appear in `Representations of Reductive Groups, in Honor of 60th Birthday of David Vogan', edited by M. Nervins and P. Trapa, published by Springe

Special unipotent representations of real classical groups: counting and reduction

Let G be a real reductive group in Harish-Chandra's class. We derive some consequences of theory of coherent continuation representations to the counting of irreducible representations of G with a given infinitesimal character and a given bound of the complex associated variety. When G is a real classical group (including the real metaplectic group), we investigate the set of special unipotent representations of G attached to Oˇ, in the sense of Arthur and Barbasch-Vogan. Here Oˇ is a nilpotent adjoint orbit in the Langlands dual of G (or the metaplectic dual of G when G is a real metaplectic group). We give a precise count for the number of special unipotent representations of G attached to Oˇ. We also reduce the problem of constructing special unipotent representations attached to Oˇ to the case when Oˇ is analytically even (equivalently for a real classical group, has good parity in the sense of Mœglin). The paper is the first in a series of two papers on the classification of special unipotent representations of real classical groups

Schubert varieties and generalizations

This contribution reviews the main results on Schubert varieties and their generalizations It covers more or less the material of the lectures at the Seminar These were partly expository introducing material needed by other lecturers In particular Section reviews classical material used in several of the other contribution

Parabolically induced representations of graded Hecke algebras

We study the representation theory of graded Hecke algebras, starting from scratch and focusing on representations that are obtained with induction from a discrete series representation of a parabolic subalgebra. We determine all intertwining operators between such parabolically induced representations, and use them to parametrize the irreducible representations.Comment: In the second version several new results have been added to prove some claims from the last page of the first version. In the third version the introduction has been extended and we determine the global dimension of a graded Hecke algebr

Derivatives for smooth representations of GL(n,R) and GL(n,C)

The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].Comment: First version of this preprint was split into 2. The proofs of two theorems which are technically involved in analytic difficulties were separated into "Twisted homology for the mirabolic nilradical" preprint. All the rest stayed in v2 of this preprint. v3: version to appear in the Israel Journal of Mathematic

On limit multiplicites of discrete series representations in spaces of automorphic forms

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46616/1/222_2005_Article_BF01388963.pd

Ladder representations of GL(n,ℚp)

In this paper, we recover certain known results about the ladder representations of GL(n, ℚp) 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