Unitary $\mathcal I$spherical representations for split $p$adic $\mathbf E_6$
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Abstract:
The determination of the Iwahorispherical unitary representations for split $p$adic groups can be reduced to the classification of unitary representations with real infinitesimal character for associated graded Hecke algebras. We determine the unitary modules with real infinitesimal character for the graded Hecke algebra of type $E_6$.References

[Al]Al D. Alvis, Induce/Restrict matrices for exceptional Weyl groups, preprint.
[Ba]B2 D. Barbasch, Unitary spherical spectrum for split classical groups, preprint, http://www.math.cornell.edu/~barbasch.
 Dan Barbasch and Dan Ciubotaru, Spherical unitary principal series, Pure Appl. Math. Q. 1 (2005), no.ย 4, Special Issue: In memory of Armand Borel., 755โ789. MR 2200999, DOI 10.4310/PAMQ.2005.v1.n4.a3 [BC2]BC1 D. Barbasch, D. Ciubotaru, Spherical unitary dual for split groups of exceptional type, preprint.
 Dan Barbasch and Allen Moy, A unitarity criterion for $p$adic groups, Invent. Math. 98 (1989), no.ย 1, 19โ37. MR 1010153, DOI 10.1007/BF01388842
 Dan Barbasch and Allen Moy, Reduction to real infinitesimal character in affine Hecke algebras, J. Amer. Math. Soc. 6 (1993), no.ย 3, 611โ635. MR 1186959, DOI 10.1090/S08940347199311869590
 Dan Barbasch and Allen Moy, Unitary spherical spectrum for $p$adic classical groups, Acta Appl. Math. 44 (1996), no.ย 12, 3โ37. Representations of Lie groups, Lie algebras and their quantum analogues. MR 1407038, DOI 10.1007/BF00116514
 Dan Barbasch and David A. Vogan Jr., Unipotent representations of complex semisimple groups, Ann. of Math. (2) 121 (1985), no.ย 1, 41โ110. MR 782556, DOI 10.2307/1971193
 Armand Borel, Admissible representations of a semisimple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233โ259. MR 444849, DOI 10.1007/BF01390139
 W. M. Beynon and N. Spaltenstein, Green functions of finite Chevalley groups of type $E_{n}$ $(n=6,\,7,\,8)$, J. Algebra 88 (1984), no.ย 2, 584โ614. MR 747534, DOI 10.1016/00218693(84)90084X
 Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
 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 WileyInterscience Publication. MR 794307
 W. Casselman, A new nonunitarity argument for $p$adic representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no.ย 3, 907โ928 (1982). MR 656064
 Dan Ciubotaru, The unitary $\Bbb I$spherical dual for split $p$adic groups of type $F_4$, Represent. Theory 9 (2005), 94โ137. MR 2123126, DOI 10.1090/S1088416505002062
 Sam Evens, The Langlands classification for graded Hecke algebras, Proc. Amer. Math. Soc. 124 (1996), no.ย 4, 1285โ1290. MR 1322921, DOI 10.1090/S0002993996032959
 J. S. Frame, The classes and representations of the groups of $27$ lines and $28$ bitangents, Ann. Mat. Pura Appl. (4) 32 (1951), 83โ119. MR 47038, DOI 10.1007/BF02417955
 David Kazhdan and George Lusztig, Proof of the DeligneLanglands conjecture for Hecke algebras, Invent. Math. 87 (1987), no.ย 1, 153โ215. MR 862716, DOI 10.1007/BF01389157
 George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no.ย 3, 599โ635. MR 991016, DOI 10.1090/S08940347198909910169
 George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes รtudes Sci. Publ. Math. 67 (1988), 145โ202. MR 972345, DOI 10.1007/BF02699129
 George Lusztig, Cuspidal local systems and graded Hecke algebras. II, Representations of groups (Banff, AB, 1994) CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, RI, 1995, pp.ย 217โ275. With errata for Part I [Inst. Hautes รtudes Sci. Publ. Math. No. 67 (1988), 145โ202; MR0972345 (90e:22029)]. MR 1357201, DOI 10.1090/S1088416502001723
 G. Lusztig, Cuspidal local systems and graded Hecke algebras. III, Represent. Theory 6 (2002), 202โ242. MR 1927954, DOI 10.1090/S1088416502001723
 Goran Muiฤ, The unitary dual of $p$adic $G_2$, Duke Math. J. 90 (1997), no.ย 3, 465โ493. MR 1480543, DOI 10.1215/S0012709497090128
 T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173โ207. MR 442103, DOI 10.1007/BF01390009
 T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no.ย 3, 279โ293. MR 491988, DOI 10.1007/BF01403165
 Marko Tadiฤ, Classification of unitary representations in irreducible representations of general linear group (nonArchimedean case), Ann. Sci. รcole Norm. Sup. (4) 19 (1986), no.ย 3, 335โ382. MR 870688, DOI 10.24033/asens.1510
 David A. Vogan Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no.ย 1, 141โ187. MR 750719, DOI 10.2307/2007074
 David A. Vogan Jr., The unitary dual of $\textrm {GL}(n)$ over an Archimedean field, Invent. Math. 83 (1986), no.ย 3, 449โ505. MR 827363, DOI 10.1007/BF01394418
Additional Information
 Dan Ciubotaru
 Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
 MR Author ID: 754534
 Email: ciubo@math.mit.edu
 Received by editor(s): November 15, 2005
 Received by editor(s) in revised form: August 23, 2006
 Published electronically: October 25, 2006
 © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.  Journal: Represent. Theory 10 (2006), 435480
 MSC (2000): Primary 22E50
 DOI: https://doi.org/10.1090/S1088416506003013
 MathSciNet review: 2266699