Duke Mathematical Journal

An exotic Deligne-Langlands correspondence for symplectic groups

Syu Kato

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Let G=Sp(2n,C) be a complex symplectic group. We introduce a (G×(C×)+1)-variety N, which we call the -exotic nilpotent cone. Then, we realize the Hecke algebra H of type Cn(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N2. This enables us to establish a Deligne-Langlands–type classification of simple H-modules under a mild assumption on parameters. As applications, we present a character formula and multiplicity formulas of H-modules

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Duke Math. J. Volume 148, Number 2 (2009), 305-371.

First available in Project Euclid: 22 May 2009

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Primary: 20G99: None of the above, but in this section


Kato, Syu. An exotic Deligne-Langlands correspondence for symplectic groups. Duke Math. J. 148 (2009), no. 2, 305--371. doi:10.1215/00127094-2009-028. https://projecteuclid.org/euclid.dmj/1242998669

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  • P. N. Achar and A. Henderson, Orbit closures in the enhanced nilpotent cone, Adv. Math. 219 (2008), 27--62.
  • H. Bass and W. J. Haboush, Linearizing certain reductive group actions, Trans. Amer. Math. Soc. 292 (1985), 463--482.
  • J. Bernstein and V. Lunts, Equivariant Sheaves and Functors, Lecture Notes in Math. 1578, Springer, Berlin, 1994.
  • A. Borel and J. De Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200--221.
  • R. W. Carter, Finite Groups of Lie type: Conjugacy Classes and Complex Characters, Pure Appl. Math. (N.Y.), Wiley, New York, 1985.
  • N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhäuser, Boston, 1997.
  • D. Ciubotaru and S. Kato, Tempered modules in exotic Deligne-Langlands correspondence, preprint,\arxiv0901.3918v1[math.RT]
  • C. De Concini, G. Lusztig, and C. Procesi, Homology of the zero-set of a nilpotent vector field on a flag manifold, J. Amer. Math. Soc. 1 (1988), 15--34.
  • J. Dadok and V. Kac, Polar representations, J. Algebra 92 (1985), 504--524.
  • N. Enomoto, Classification of the irreducible representations of affine Hecke algebras of type $B_2$ with unequal parameters, J. Math. Kyoto Univ. 46 (2006), 259--273.
  • —, A quiver construction of symmetric crystals, to appear in Int. Math. Res. Notices, preprint,\arxiv0806.3615v2[math.RT]
  • M. Finkelberg, V. Ginzburg, and R. Travkin, Mirabolic affine Grassmannian and character sheaves, to appear in Selecta Math., preprint,\arxiv0802.1652v2[math.AG]
  • V. Ginzburg, ``Geometric methods in representation theory of Hecke algebras and quantum groups'' in Representation Theories and Algebraic Geometry (Montreal, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 514, Kluwer, Dordrecht, 1998, 127--183.
  • R. Hartshorne, Algebraic Geometry, Grad. Texts in Math. 52, Springer, New York, 1977.
  • J.-I. Igusa, ``Geometry of absolutely admissible representations'' in Number Theory, Algebraic Geometry and Commutative Algebra, in Honor of Yasuo Akizuki, Kinokuniya, Tokyo, 1973, 373--452.
  • R. Joshua, Modules over convolution algebras from derived categories, I, J. Algebra 203 (1998), 385--446.
  • S. Kato, An exotic Springer correspondence for symplectic groups, preprint,\arxivmath/0607478v2[math.RT]
  • —, Deformations of nilpotent cones and Springer correspondences, to appear in Amer. J. Math., preprint,\arxiv0801.3707v3[math.RT]
  • D. Kazhdan and G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153--215.
  • G. Lusztig, Cuspidal local systems and graded Hecke algebras, I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145--202.
  • —, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599--635.
  • —, Classification of unipotent representations of simple p-adic groups. Internat. Math. Res. Notices 1995, no. 11, 517--589.
  • —, ``Cuspidal local systems and graded Hecke algebras, II'' with errata for part I, in Representations of Groups (Banff, Canada, 1994), CMS Conf. Proc. 16, Amer. Math. Soc., Providence, 1995, 217--275.
  • —, Cuspidal local systems and graded Hecke algebras, III, Represent. Theory 6 (2002), 202--242.
  • —, Hecke Algebras with Unequal Parameters, CRM Monogr. Ser. 18, Amer. Math. Soc., Providence, 2003.
  • I. G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts Math. 157, Cambridge Univ. Press, Cambridge, 2003.
  • T. Ohta, The singularities of the closures of nilpotent orbits in certain symmetric pairs, Tôhoku Math. J. 38 (1986), 441--468.
  • E. Opdam and M. Solleveld, Homological algebra for affine Hecke algebras, Adv. Math. 220 (2009), 1549--1601.
  • —, Discrete series characters for affine Hecke algebras and their formal degrees, to appear in Acta Math., preprint,\arxiv0804.0026v1[math.RT]
  • V. L. Popov, The cone of Hilbert null forms, Proc. Steklov Inst. Math. 2003, no. 2, 177--194.
  • A. Ram, ``Representations of rank two affine Hecke algebras'' in Advances in Algebra and Geometry (Hyderabad, India, 2001), Hindustan Book Agency, New Delhi, 2003, 57-91.
  • M. Saito, Modules de Hodge polarisables, Publ. Res. Inst. Math. Sci. 24 (1988), 849--995.
  • G. W. Schwarz, Representations of simple groups with a free module of covariants, Invent. Math. 50 (1978/79), 1--12.
  • J. Sekiguchi, The nilpotent subvariety of the vector space associated to a symmetric pair, Publ. Res. Inst. Math. Sci. 20 (1984), 155--212.
  • M. Solleveld, Periodic cyclic homology of reductive p-adic groups, preprint\arxiv0710.5815v1[math.KT]
  • N. Spaltenstein, Classes unipotentes et sous groupes de Borel, Lecture Notes in Math. 946, Springer, Berlin, 1982.
  • T. A. Springer, The exotic nilcone of a symplectic group, preprint, to appear in J. Algebra.
  • R. W. Thomason, Lefschetz-Riemann-Roch theorem and coherent trace formula, Invent. Math. 85 (1986), 515--543.
  • R. Travkin, Mirabolic Robinson-Schensted-Knuth correspondence, to appear in Selecta Math., preprint,\arxiv08020 1651v2[math.AG]