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Double affine Hecke algebras and 2-dimensional local fields

Author: M. Kapranov
Journal: J. Amer. Math. Soc. 14 (2001), 239-262
MSC (2000): Primary 20C08; Secondary 20G25
Published electronically: September 25, 2000
MathSciNet review: 1800352
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We give an interpretation of the double affine Hecke algebra of Cherednik as a (suitably regularized) algebra of double cosets of a group $G$ by a subgroup $\mathcal F$, extending the well-known interpretations of the finite and affine Hecke algebras. In this interpretation, $G$ consists of $K$-points of a simple algebraic group, where $K$ is a 2-dimensional local field such as $\mathbf Q_p((t))$ or $F_q((t_1))((t_2))$, and $\mathcal F$ is a certain analog of the Iwahori subgroup.

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  • [AGV] M. Artin, A. Grothendieck, J.-L. Verdier, SGA4, Theorie des Topos et Cohomologie Etale des Schemas, t.1, Lecture Notes in Math. 269, Springer-Verlag, 1972. MR 50:7130
  • [AM] M. Artin, B. Mazur, Etale Homotopy, Lecture Notes in Math. 100, Springer-Verlag, 1969. MR 88a:14024
  • [B] K. Brown, Buildings, Springer-Verlag, 1989. MR 90e:20001
  • [Cas] W. Casselman, Unramified principal series for p-adic groups I. The spherical function, Compositio Math. 40 (1980), 387-406. MR 83a:22018
  • [Ch] I. Cherednik, Double affine Hecke algebras and Macdonald's conjectures, Ann. Math. 141 (1995), 191-216. MR 96m:33010
  • [CK] N. Chriss, K. Khuri-Makdisi, On the Iwahori-Hecke algebra of a p-adic group, Int. Math. Res. Notices, 1998, no.2, 85-100. MR 99c:22023
  • [Dr] V.G. Drinfeld, Two-dimensional $l$-adic representations of the fundamental group of a curve over a finite field and automorphic forms on $GL(2)$, Amer. J. Math. 105 (1983), 85-114. MR 86i:11066
  • [FP] T. Fimmel, A.N. Parshin, Introduction to Higher Adelic Theory, book in preparation.
  • [GZ] P. Gabriel, M. Zisman, Calculus of Fractions and Homotopy Theory (Ergebnisse der Math. 35), Springer-Verlag, 1967. MR 35:1019
  • [GG] H. Garland, I. Grojnowski, Affine Hecke algebras associated to Kac-Moody groups, preprint q-alg/9508019.
  • [GGP] I.M. Gelfand, M.I. Graev, I.I. Piatetski-Shapiro, Representation theory and automorphic functions, Academic Press, 1969. MR 38:2093
  • [GKV] V. Ginzburg, M. Kapranov, E. Vasserot, Residue construction of Hecke algebras, Adv. in Math. 128 (1997), 1-19. MR 98k:20074
  • [Gra] J.W. Gray, Formal Category Theory, Lecture Notes in Math. 391, Springer-Verlag, 1974. MR 51:8207
  • [Kac] V. Kac, Infinite-dimensional Lie algebras, Cambridge Univ. Press, 1985. MR 87e:17023
  • [Kat] K. Kato, The existence theorem for higher local class field theory, preprint M/80/43, IHES, 1980.
  • [KL] D. Kazhdan, G. Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), 153-215. MR 88d:11121
  • [Lu1] G. Lusztig, Singularities, character formula and $q$-analog of weight multiplicity, Asterisque 101-102 (1983), 208-222. MR 85m:17005
  • [Lu2] G. Lusztig, Intersection cohomology methods in representation theory, Proc. ICM-90, vol.1, pp. 155-174, Math. Soc. Japan, Tokyo, 1991. MR 92m:20034
  • [Mat] H.Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. ENS 2 (1969), 1-62. MR 39:1566
  • [Mil] J. Milnor, Introduction to Algebraic K-theory, Princeton Univ. Press, 1971. MR 50:2304
  • [Pa1] A.N. Parshin, On the arithmetic of 2-dimensional schemes. I, Repartitions and residues, Russian Math. Izv. 40 (1976), 736-773.
  • [Pa2] A.N. Parshin, Vector bundles and arithmetic groups I: The higher Bruhat-Tits tree, Proc. Steklov Inst. Math. 208 (1995), 212-233, preprint alg-geom/9605001. MR 2000j:20090
  • [PS] A. Pressley, G. Segal, Loop Groups, Clarendon Press, Oxford, 1986. MR 88i:22049
  • [Th] R.W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Phil. Soc. 85 (1979), 91-109. MR 80b:18015

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Additional Information

M. Kapranov
Affiliation: Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, Canada M5S 3G3

Received by editor(s): June 8, 1999
Received by editor(s) in revised form: March 16, 2000, and July 25, 2000
Published electronically: September 25, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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