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These were defined and studied by Lusztig and Vogan in [10]. In order to carry out the computation, we introduce new class of twisted parameters, and study the Hecke algebra action in the resulting basis. We work in the equivalent setting of graded Hecke algebra modules. We make a connection between the semisimplicity of Hecke algebra modules, unitarity with respect to a certain hermitian form, and ladder representations.

Let k be a field, let G be a reductive group, and let V be a linear representation of G. We illustrate the various scenarios that can occur with some recent examples of arithmetic interest. 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. This continues the study begun in [7], where the locus of rationally smooth points was determined. In this paper, we give explicit descriptions of the centers and cocenters of 0-Hecke algebras associated to finite Coxeter groups.

A few problems and conjectures are proposed for further investigations. Our primary concern is with representations of reductive Lie groups, which involve both algebraic and analytic approaches. We divide branching problems into three stages: A abstract features of the restriction; B branching laws irreducible decompositions of the restriction ; and C construction of symmetry breaking operators on geometric models.

The aim of this article is to give new perspectives on the subjects, to explain the methods based on some recent progress, and to raise some conjectures and open questions. This paper connects results on Amitsur—Levitski identities for simple Lie algebras, ideals in Borel subalgebras, commutative Lie subalgebras in simple Lie algebras, filtration of sheets, and recent work with Nolan Wallach on the variety of singular elements in a complex semisimple Lie algebra.

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Let G be a connected reductive group over an algebraically closed field. We define a decomposition of G into finitely many strata such that each stratum is a union of conjugacy classes of fixed dimension; the strata are indexed purely in terms of the Weyl group and the indexing set is independent of the characteristic. We classify all of the pre-canonical structures which arise in this manner, and explain the relationships between their resulting canonical bases. Toggle navigation. Search Video Library.

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