Classification of one -type representations

Authors:
Dan Barbasch and Allen Moy

Journal:
Trans. Amer. Math. Soc. **351** (1999), 4245-4261

MSC (1991):
Primary 22E50; Secondary 20G05

Published electronically:
June 29, 1999

MathSciNet review:
1473430

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Abstract | References | Similar Articles | Additional Information

Abstract: Suppose is a simple reductive -adic group with Weyl group . We give a classification of the irreducible representations of which can be extended to real hermitian representations of the associated graded Hecke algebra . Such representations correspond to unitary representations of which have a small spectrum when restricted to an Iwahori subgroup.

**[A]**D. Alvis,*Induce/Restrict matrices for exceptional Weyl groups*, preprint.**[BM1]**Dan Barbasch and Allen Moy,*A unitarity criterion for 𝑝-adic groups*, Invent. Math.**98**(1989), no. 1, 19–37. MR**1010153**, 10.1007/BF01388842**[BM2]**Dan Barbasch and Allen Moy,*Reduction to real infinitesimal character in affine Hecke algebras*, J. Amer. Math. Soc.**6**(1993), no. 3, 611–635. MR**1186959**, 10.1090/S0894-0347-1993-1186959-0**[BS]**W.M. Benyon and N. Spaltenstein,*The computation of Green's functions of finite Chevalley groups of type ()*, The University of Warwick, Computer Centre, report no 23.**[C]**Roger W. Carter,*Finite groups of Lie type*, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1985. Conjugacy classes and complex characters; A Wiley-Interscience Publication. MR**794307****[G]***GAP*, Lehrstuhl D fuer Mathematik, RWTH Aachen http://www.ccs.neu.edu/mirrors/GAP.**[J]**Hervé Jacquet,*On the residual spectrum of 𝐺𝐿(𝑛)*, Lie group representations, II (College Park, Md., 1982/1983) Lecture Notes in Math., vol. 1041, Springer, Berlin, 1984, pp. 185–208. MR**748508**, 10.1007/BFb0073148**[KL]**David Kazhdan and George Lusztig,*Proof of the Deligne-Langlands conjecture for Hecke algebras*, Invent. Math.**87**(1987), no. 1, 153–215. MR**862716**, 10.1007/BF01389157**[Ls1]**George Lusztig,*Affine Hecke algebras and their graded version*, J. Amer. Math. Soc.**2**(1989), no. 3, 599–635. MR**991016**, 10.1090/S0894-0347-1989-0991016-9**[Ls2]**G. Lusztig,*A class of irreducible representations of a Weyl group*, Nederl. Akad. Wetensch. Indag. Math.**41**(1979), no. 3, 323–335. MR**546372****[MW]**C. Mœglin and J.-L. Waldspurger,*Le spectre résiduel de 𝐺𝐿(𝑛)*, Ann. Sci. École Norm. Sup. (4)**22**(1989), no. 4, 605–674 (French). MR**1026752****[S]**Birgit Speh,*Unitary representations of 𝐺𝑙(𝑛,𝑅) with nontrivial (𝔤,𝔎)-cohomology*, Invent. Math.**71**(1983), no. 3, 443–465. MR**695900**, 10.1007/BF02095987**[T]**Marko Tadić,*Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case)*, Ann. Sci. École Norm. Sup. (4)**19**(1986), no. 3, 335–382. MR**870688**

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Additional Information

**Dan Barbasch**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Email:
barbasch@math.cornell.edu

**Allen Moy**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Email:
moy@math.loa.umich.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-99-02171-6

Received by editor(s):
July 1, 1996

Received by editor(s) in revised form:
June 23, 1997

Published electronically:
June 29, 1999

Additional Notes:
The authors were supported in part by NSF grants DMS–9401176 and DMS–9500973

Article copyright:
© Copyright 1999
American Mathematical Society