Unitary ℐ-spherical representations for split 𝓅-adic 𝐄₆

Ciubotaru, Dan

doi:10.1090/S1088-4165-06-00301-3

Unitary $\mathcal I$-spherical representations for split $p$-adic $\mathbf E_6$
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by Dan Ciubotaru

Represent. Theory 10 (2006), 435-480

DOI: https://doi.org/10.1090/S1088-4165-06-00301-3

Published electronically: October 25, 2006

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Abstract:

The determination of the Iwahori-spherical 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

Induce/Restrict matrices for exceptional Weyl groups

Unitary spherical spectrum for split classical groups

http://www.math.cornell.edu/~barbasch

Dan Barbasch and Dan Ciubotaru, Spherical unitary principal series, Pure Appl. Math. Q. 1 (2005), no. 4, Special Issue: In memory of Armand Borel., 755–789. MR 2200999, DOI 10.4310/PAMQ.2005.v1.n4.a3

Spherical unitary dual for split groups of exceptional type

Dan Barbasch and Allen Moy, A unitarity criterion for $p$-adic groups, Invent. Math. 98 (1989), no. 1, 19–37. 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. 3, 611–635. MR 1186959, DOI 10.1090/S0894-0347-1993-1186959-0
Dan Barbasch and Allen Moy, Unitary spherical spectrum for $p$-adic classical groups, Acta Appl. Math. 44 (1996), no. 1-2, 3–37. 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. 1, 41–110. MR 782556, DOI 10.2307/1971193
Armand Borel, Admissible representations of a semi-simple group over a local field with vectors fixed under an Iwahori subgroup, Invent. Math. 35 (1976), 233–259. 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. 2, 584–614. MR 747534, DOI 10.1016/0021-8693(84)90084-X
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 Wiley-Interscience Publication. MR 794307
W. Casselman, A new nonunitarity argument for $p$-adic representations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 907–928 (1982). MR 656064
Dan Ciubotaru, The unitary $\Bbb I$-spherical dual for split $p$-adic groups of type $F_4$, Represent. Theory 9 (2005), 94–137. MR 2123126, DOI 10.1090/S1088-4165-05-00206-2
Sam Evens, The Langlands classification for graded Hecke algebras, Proc. Amer. Math. Soc. 124 (1996), no. 4, 1285–1290. MR 1322921, DOI 10.1090/S0002-9939-96-03295-9
J. S. Frame, The classes and representations of the groups of $27$ lines and $28$ bitangents, Ann. Mat. Pura Appl. (4) 32 (1951), 83–119. MR 47038, DOI 10.1007/BF02417955
David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, DOI 10.1007/BF01389157
George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
George Lusztig, Cuspidal local systems and graded Hecke algebras. I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145–202. 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. 217–275. With errata for Part I [Inst. Hautes Études Sci. Publ. Math. No. 67 (1988), 145–202; MR0972345 (90e:22029)]. MR 1357201, DOI 10.1090/S1088-4165-02-00172-3
G. Lusztig, Cuspidal local systems and graded Hecke algebras. III, Represent. Theory 6 (2002), 202–242. MR 1927954, DOI 10.1090/S1088-4165-02-00172-3
Goran Muić, The unitary dual of $p$-adic $G_2$, Duke Math. J. 90 (1997), no. 3, 465–493. MR 1480543, DOI 10.1215/S0012-7094-97-09012-8
T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207. MR 442103, DOI 10.1007/BF01390009
T. A. Springer, A construction of representations of Weyl groups, Invent. Math. 44 (1978), no. 3, 279–293. MR 491988, DOI 10.1007/BF01403165
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, DOI 10.24033/asens.1510
David A. Vogan Jr., Unitarizability of certain series of representations, Ann. of Math. (2) 120 (1984), no. 1, 141–187. 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. 3, 449–505. MR 827363, DOI 10.1007/BF01394418