×

zbMATH — the first resource for mathematics

Characters of Springer representations on elliptic conjugacy classes. (English) Zbl 1260.22012
For a Weyl group \(W,\) the authors investigate simple closed formulas (valid on elliptic conjugacy classes) for the character of the representation of \(W\) in the homology of a Springer fiber. They also give a formula (valid again on elliptic conjugacy classes) of the \(W\)-character of an irreducible discrete series representation with real central character of a graded affine Hecke algebra with arbitrary parameters. In both cases, the Pin double cover of \(W\) and the Dirac operator for graded affine Hecke algebras play key roles.

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
PDF BibTeX XML Cite
Full Text: DOI Euclid arXiv
References:
[1] D. Barbasch, D. Ciubotaru, and P. Trapa, The Dirac operator for graded affine Hecke algebras , to appear in Acta Math., preprint, [math.RT]. 1006.3822v1 · Zbl 1276.20004 · doi:10.1007/s11511-012-0085-3 · arxiv.org
[2] A. Borel and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups , Ann. of Math. Stud. 94 , Princeton Univ. Press, Princeton, 1980. · Zbl 0443.22010
[3] W. Borho and R. MacPherson, Représentations des groupes de Weyl et homologie d’intersection pour les variétés nilpotentes , C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 707-710. · Zbl 0467.20036
[4] R. Carter, Conjugacy classes in the Weyl group , Compositio Math. 25 (1972), 1-59. · Zbl 0254.17005 · numdam:CM_1972__25_1_1_0 · eudml:89111
[5] D. Ciubotaru, On unitary unipotent representations of \(p\)-adic groups and affine Hecke algebras with unequal parameters , Represent. Theory 12 (2008), 453-498. · Zbl 1157.22009 · doi:10.1090/S1088-4165-08-00338-5
[6] D. Ciubotaru, Spin representations of Weyl groups and the Springer correspondence , J. Reine Angew. Math. 671 (2012), 199-222. · Zbl 1272.17012
[7] D. Ciubotaru, M. Kato, and S. Kato, On characters and formal degrees of discrete series of classical affine Hecke algebras , Invent. Math. 187 (2012), 589-635. · Zbl 1270.20002 · doi:10.1007/s00222-011-0338-3
[8] D. Ciubotaru, E. M. Opdam, and P. E. Trapa, Algebraic and analytic Dirac induction for graded affine Hecke algebras , preprint, [math.RT] 1201.2130v2 · Zbl 1362.20004 · arxiv.org
[9] D. H. Collingwood and W. M. McGovern, Nilpotent Orbits in Semisimple Lie Algebras , Van Nostrand Reinhold Math. Ser., Van Nostrand Reinhold, New York, 1993. · Zbl 0972.17008
[10] P. Gunnels and E. Sommers, A characterization of Dynkin elements , Math. Res. Letters 10 (2003), 363-373. · Zbl 1139.17302 · doi:10.4310/MRL.2003.v10.n3.a6
[11] D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture , Invent. Math. 87 (1987), 153-215. · Zbl 0613.22004 · doi:10.1007/BF01389157 · eudml:143417
[12] G. Lusztig, Character sheaves, V , Adv. Math. 61 (1986), 103-155. · Zbl 0602.20036 · doi:10.1016/0001-8708(86)90071-X
[13] G. Lusztig, Cuspidal local systems and graded algebras, I , Publ. Math. Inst. Hautes Études Sci. 67 (1988), 145-202. · Zbl 0699.22026 · doi:10.1007/BF02699129 · numdam:PMIHES_1988__67__145_0 · eudml:104030
[14] G. Lusztig, Affine Hecke algebras and their graded versions , J. Amer. Math. Soc. 2 (1989), 599-635. · Zbl 0715.22020 · doi:10.2307/1990945
[15] G. Lusztig, “Cuspidal local systems and graded algebras, II” in Representations of Groups (Banff, AB, 1994) , Amer. Math. Soc., Providence, 1995, 217-275. · Zbl 0841.22013
[16] E. Opdam, On the spectral decomposition of affine Hecke algebras , J. Inst. Math. Jussieu 3 (2004), 531-648. · Zbl 1102.22009 · doi:10.1017/S1474748004000155
[17] E. Opdam and M. Solleveld, Homological algebra for affine Hecke algebras , Adv. in Math. 220 (2009), 1549-1601. · Zbl 1195.20004 · doi:10.1016/j.aim.2008.11.002
[18] M. Reeder, Euler-Poincaré pairings and elliptic representations of Weyl groups and \(p\)-adic groups , Compositio Math. 129 (2001), 149-181. · Zbl 1037.22039 · doi:10.1023/A:1014539331377
[19] P. Schneider and U. Stuhler, Representation theory and sheaves on the Bruhat-Tits building , Publ. Math. Inst. Hautes Études Sci. 85 (1997), 97-191. · Zbl 0892.22012 · doi:10.1007/BF02699536 · numdam:PMIHES_1997__85__97_0 · eudml:104122
[20] T. Shoji, On the Green polynomials of classical groups , Invent. Math. 74 (1983), 239-267. · Zbl 0525.20027 · doi:10.1007/BF01394315 · eudml:143071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.