## 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.## References

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## Bibliographic Information

**Cathy Kriloff**- Affiliation: Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085
- MR Author ID: 630044
- ORCID: 0000-0003-2863-6724
- Email: krilcath@isu.edu
**Arun Ram**- Affiliation: Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
- MR Author ID: 316170
- Email: ram@math.wisc.edu
- Received by editor(s): May 15, 2001
- Received by editor(s) in revised form: December 21, 2001, and January 23, 2002
- Published electronically: May 2, 2002
- Additional Notes: Research of the first author supported in part by an NSF-AWM Mentoring Travel Grant

Research of the second author supported in part by National Security Agency grant MDA904-01-1-0032 and EPSRC Grant GR K99015 - © Copyright 2002 American Mathematical Society
- Journal: Represent. Theory
**6**(2002), 31-69 - MSC (2000): Primary 20C08; Secondary 16G99
- DOI: https://doi.org/10.1090/S1088-4165-02-00160-7
- MathSciNet review: 1915086