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 Representation Theory
Representation Theory
ISSN 1088-4165

     

On unitary unipotent representations of $ p$-adic groups and affine Hecke algebras with unequal parameters

Author(s): Dan Ciubotaru
Journal: Represent. Theory 12 (2008), 453-498.
MSC (2000): Primary 22E50
Posted: December 15, 2008
MathSciNet review: 2465803
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Abstract | References | Similar articles | Additional information

Abstract: We determine the unitary dual of the geometric graded Hecke algebras with unequal parameters which appear in Lusztig's classification of unipotent representations for exceptional $ p$-adic groups. The largest such algebra is of type $ F_4.$ Via the Barbasch-Moy correspondence of unitarity applied to this setting, this is equivalent to the identification of the corresponding unitary unipotent representations with real central character of the $ p$-adic groups. In order for this correspondence to be applicable here, we show (following Lusztig's geometric classification, and Barbasch and Moy's original argument) that the set of tempered modules with real central character for a geometric graded Hecke algebra is linearly independent when restricted to the Weyl group.

References:

[Al]
D. Alvis Induce/Restrict matrices for exceptional Weyl groups, preprint.

[Ba]
D. Barbasch Unitary spherical spectrum for split classical groups, preprint, arXiv:math/0609828.

[BC1]
D. Barbasch, D. Ciubotaru, Whittaker unitary dual of affine graded Hecke algebras of type $ E$, preprint, arXiv:0708.4036.

[BC2]
-, Spherical unitary principal series, Pure Appl. Math. Q. 1 (2005), no. 4, 755-789. MR 2200999 (2006m:22023)

[BM1]
D. Barbasch, A. Moy, A unitarity criterion for $ p$-adic groups, Invent. Math. 98, 1989, 19-38. MR 1010153 (90m:22038)

[BM2]
-, Reduction to real infinitesimal character in affine Hecke algebras, J. Amer. Math. Soc. 6(3), 1993, 611-635. MR 1186959 (93k:22015)

[BM3]
-, Unitary spherical spectrum for $ p$-adic classical groups, Acta Appl. Math. 44, 1996, 1-37. MR 1407038 (98k:22067)

[BM4]
-, Whittaker models with an Iwahori fixed vector, Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993), 101-105, Contemp. Math., 177, Amer. Math. Soc., Providence, RI, 1994. MR 1303602 (95j:22024)

[BS]
W. Beynon, N. Spaltenstein, Green functions of finite Chevalley groups of type $ E_n (n=6,7,8)$, J. Algebra 88 (1984), 584-614. MR 747534 (85k:20136)

[BW]
A. Borel, N. Wallach, Continuous cohomology, discrete subgroups and representations of reductive groups, Annals of Mathematics Studies, vol. 94, Princeton University Press, Princeton, 1980. MR 554917 (83c:22018)

[Ca]
R. Carter, Finite groups of Lie type, Wiley-Interscience, New York, 1985. MR 794307 (87d:20060)

[Ci1]
D. Ciubotaru, The unitary $ \mathbb{I}$-spherical dual for split $ p$-adic groups of type $ F_4$, Represent. Theory, vol. 9, 2005, 94-137. MR 2123126 (2005k:22022)

[Ci2]
-, Unitary $ I$-spherical representations for split $ p$-adic $ E_6$, Represent. Theory 10 (2006), 435-480. MR 2266699 (2008d:22017)

[Ev]
S. Evens, The Langlands classification for graded Hecke algebras, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1285-1290. MR 1322921 (96g:22022)

[Ka]
S. Kato, An exotic Deligne-Langlands correspondence for symplectic groups, preprint, arXiv:math/0601155.

[KL]
D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87, 1987, 153-215. MR 862716 (88d:11121)

[Kn]
A.W. Knapp, Representation theory of semisimple groups. An overview based on examples, Princeton Mathematical Series, 36, Princeton University Press, 1986. MR 855239 (87j:22022)

[KR]
C. Kriloff, A. Ram, Representations of graded Hecke algebras, Represent. Theory 6 (2002), 31-69. MR 1915086 (2003h:20013)

[LS]
L. Lambe, B. Srinivasan, A computation of Green functions for some classical groups, Comm. Algebra 18 (1990), no. 10, 3507-3545. MR 1063992 (91i:20041)

[Lu1]
G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2, 1989, 599-635. MR 991016 (90e:16049)

[Lu2]
-, Cuspidal local systems and graded algebras I, Publ. Math. de l'IHES 67, 1988, 145-202. MR 972345 (90e:22029)

[Lu3]
-, Cuspidal local systems and graded algebras II, Representations of groups (Banff, AB, 1994), Amer. Math. Soc., Providence, 1995, 217-275. MR 1357201 (96m:22038)

[Lu4]
-, Cuspidal local systems and graded algebras III, Represent. Theory 6 (2002), 202-242. MR 1927954 (2004k:20010)

[Lu5]
-, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205-272. MR 732546 (86d:20050)

[Lu6]
-, Classification of unipotent representations of simple $ p$-adic groups, Internat. Math. Res. Notices 1995, no. 11, 517-589. MR 1369407 (98b:22034)

[Lu7]
-, Character sheaves V, Adv. in Math. 61 (1986), 103-155. MR 849848 (87m:20118c)

[Mu]
G. Muić, The unitary dual of $ p$-adic $ G\sb 2$, Duke Math. J. 90 (1997), no. 3, 465-493. MR 1480543 (98k:22073)

[Op1]
E. Opdam, On the spectral decomposition of affine Hecke algebras, J. Inst. Math. Jussieu 3 (2004), no. 4, 531-648. MR 2094450 (2005i:20008)

[Op2]
-, A generating function for the trace of the Iwahori-Hecke algebra, Progr. Math., 210, Birkhäuser Boston, 2003, 301-323. MR 1985730 (2004f:20013)

[OS]
E. Opdam, M. Solleveld, Discrete series characters for affine Hecke algebras and their formal degrees, preprint, arXiv:0804.0026.

[Re1]
M. Reeder, Formal degrees and $ L$-packets of unipotent discrete series representations of exceptional $ p$-adic groups, J. Reine Angew. Math. 520 (2000), 37-93. MR 1748271 (2001k:22039)

[Re2]
-, Whittaker models and unipotent representations of $ p$-adic groups, Math. Ann. 308, 587-592 (1997). MR 1464911 (98h:22022)

[Sh1]
T. Shoji, On the Green polynomials of classical groups, Invent. Math. 74 (1983), 239-267. MR 723216 (85f:20032)

[Sh2]
-, On the Green polynomials of a Chevalley group of type $ F_4$, Commun. Algebra 10 (1982), 505-543. MR 647835 (83d:20030)

[Spa1]
N. Spaltenstein, On the generalized Springer correspondence for exceptional groups, Algebraic groups and related topics (Kyoto/Nagoya, 1983), 317-338, Adv. Stud. Pure Math., 6, North-Holland, Amsterdam, 1985. MR 803340 (87g:20072b)

[Spa2]
-, Classes unipotentes et sous-groupes de Borel, Lecture Notes in Mathematics, 946, Springer-Verlag, Berlin, New York, 1982. MR 672610 (84a:14024)

[Spr1]
T.A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279-293. MR 0491988 (58:11154)

[Spr2]
-, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173-207. MR 0442103 (56:491)

[Ta]
M. 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 (88b:22021)

[Ti]
J. Tits, Reductive groups over local fields, Proceedings of Symposia in Pure Math., vol. 33 (1979), 29-69. MR 546588 (80h:20064)

[Vo1]
D. A. Vogan, Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141-187. MR 750719 (86h:22028)

[Vo2]
-, Representations of real reductive Lie groups, Progress in Mathematics, 15, Birkhäuser, Boston, MA, 1981. MR 632407 (83c:22022)

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Additional Information:

Dan Ciubotaru
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112
Email: ciubo@math.utah.edu

DOI: 10.1090/S1088-4165-08-00338-5
PII: S 1088-4165(08)00338-5
Received by editor(s): January 31, 2007
Posted: December 15, 2008
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




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