Representations of graded Hecke algebras

Kriloff, Cathy; Ram, Arun

doi:10.1090/S1088-4165-02-00160-7

Representations of graded Hecke algebras
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by Cathy Kriloff and Arun Ram

Represent. Theory 6 (2002), 31-69

DOI: https://doi.org/10.1090/S1088-4165-02-00160-7

Published electronically: May 2, 2002

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Abstract:

Representations of affine and graded Hecke algebras associated to Weyl groups play an important role in the Langlands correspondence for the admissible representations of a reductive $p$-adic group. We work in the general setting of a graded Hecke algebra associated to any real reflection group with arbitrary parameters. In this setting we provide a classification of all irreducible representations of graded Hecke algebras associated to dihedral groups. Dimensions of generalized weight spaces, Langlands parameters, and a Springer-type correspondence are included in the classification. We also give an explicit construction of all irreducible calibrated representations (those possessing a simultaneous eigenbasis for the commutative subalgebra) of a general graded Hecke algebra. While most of the techniques used have appeared previously in various contexts, we include a complete and streamlined exposition of all necessary results, including the Langlands classification of irreducible representations and the irreducibility criterion for principal series representations.

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