Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


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.

References [Enhancements On Off] (What's this?)

  • [AGV] Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin-New York, 1972 (French). Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4); Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. MR 0354652
  • [AM] M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, vol. 100, Springer-Verlag, Berlin, 1986. Reprint of the 1969 original. MR 883959
  • [B] Kenneth S. Brown, Buildings, Springer-Verlag, New York, 1989. MR 969123
  • [Cas] W. Casselman, The unramified principal series of 𝔭-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
  • [Ch] Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, 10.2307/2118632
  • [CK] Neil Chriss and Kamal Khuri-Makdisi, On the Iwahori-Hecke algebra of a 𝑝-adic group, Internat. Math. Res. Notices 2 (1998), 85–100. MR 1604812, 10.1155/S1073792898000087
  • [Dr] V. G. Drinfel′d, Two-dimensional 𝑙-adic representations of the Galois group of a global field of characteristic 𝑝 and automorphic forms on 𝐺𝐿(2), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 134 (1984), 138–156 (Russian, with English summary). Automorphic functions and number theory, II. MR 741857
  • [FP] T. Fimmel, A.N. Parshin, Introduction to Higher Adelic Theory, book in preparation.
  • [GZ] P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 0210125
  • [GG] H. Garland, I. Grojnowski, Affine Hecke algebras associated to Kac-Moody groups, preprint q-alg/9508019.
  • [GGP] I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Translated from the Russian by K. A. Hirsch, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. MR 0233772
  • [GKV] Victor Ginzburg, Mikhail Kapranov, and Eric Vasserot, Residue construction of Hecke algebras, Adv. Math. 128 (1997), no. 1, 1–19. MR 1451416, 10.1006/aima.1997.1620
  • [Gra] John W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics, Vol. 391, Springer-Verlag, Berlin-New York, 1974. MR 0371990
  • [Kac] V. S. Retakh, Massey operations in Lie superalgebras and differentials of the Quillen spectral sequence, Colloq. Math. 50 (1985), no. 1, 81–94 (Russian). MR 818089
  • [Kat] K. Kato, The existence theorem for higher local class field theory, preprint M/80/43, IHES, 1980.
  • [KL] David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, 10.1007/BF01389157
  • [Lu1] George Lusztig, Singularities, character formulas, and a 𝑞-analog of weight multiplicities, Analysis and topology on singular spaces, II, III (Luminy, 1981) Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 208–229. MR 737932
  • [Lu2] George Lusztig, Intersection cohomology methods in representation theory, ICM-90, Mathematical Society of Japan, Tokyo; distributed outside Asia by the American Mathematical Society, Providence, RI, 1990. A plenary address presented at the International Congress of Mathematicians held in Kyoto, August 1990. MR 1127427
  • [Mat] Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 0240214
  • [Mil] John Milnor, Introduction to algebraic 𝐾-theory, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. Annals of Mathematics Studies, No. 72. MR 0349811
  • [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, Trudy Mat. Inst. Steklov. 208 (1995), no. Teor. Chisel, Algebra i Algebr. Geom., 240–265 (Russian). Dedicated to Academician Igor′ Rostislavovich Shafarevich on the occasion of his seventieth birthday (Russian). MR 1730268
  • [PS] Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587
  • [Th] R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91–109. MR 510404, 10.1017/S0305004100055535

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 20C08, 20G25

Retrieve articles in all journals with MSC (2000): 20C08, 20G25

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