Double affine Hecke algebras and 2-dimensional local fields

Kapranov, M.

doi:10.1090/S0894-0347-00-00354-4

Double affine Hecke algebras and 2-dimensional local fields
HTML articles powered by AMS MathViewer

by M. Kapranov;

J. Amer. Math. Soc. 14 (2001), 239-262

DOI: https://doi.org/10.1090/S0894-0347-00-00354-4

Published electronically: September 25, 2000

PDF | Request permission

Abstract:

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

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 354652
M. Artin and B. Mazur, Etale homotopy, Lecture Notes in Mathematics, vol. 100, Springer-Verlag, Berlin, 1986. Reprint of the 1969 original. MR 883959
Kenneth S. Brown, Buildings, Springer-Verlag, New York, 1989. MR 969123, DOI 10.1007/978-1-4612-1019-1
W. Casselman, The unramified principal series of ${\mathfrak {p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
Ivan Cherednik, Double affine Hecke algebras and Macdonald’s conjectures, Ann. of Math. (2) 141 (1995), no. 1, 191–216. MR 1314036, DOI 10.2307/2118632
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, DOI 10.1155/S1073792898000087
V. G. Drinfel′d, Two-dimensional $l$-adic representations of the Galois group of a global field of characteristic $p$ and automorphic forms on $\textrm {GL}(2)$, 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
P. Gabriel and M. Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 35, Springer-Verlag New York, Inc., New York, 1967. MR 210125, DOI 10.1007/978-3-642-85844-4
I. M. Gel′fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, W. B. Saunders Co., Philadelphia, Pa.-London-Toronto, Ont., 1969. Translated from the Russian by K. A. Hirsch. MR 233772
Victor Ginzburg, Mikhail Kapranov, and Eric Vasserot, Residue construction of Hecke algebras, Adv. Math. 128 (1997), no. 1, 1–19. MR 1451416, DOI 10.1006/aima.1997.1620
John W. Gray, Formal category theory: adjointness for $2$-categories, Lecture Notes in Mathematics, Vol. 391, Springer-Verlag, Berlin-New York, 1974. MR 371990, DOI 10.1007/BFb0061280
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
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, 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
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
Hideya Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés, Ann. Sci. École Norm. Sup. (4) 2 (1969), 1–62 (French). MR 240214, DOI 10.24033/asens.1174
John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1971. MR 349811

Russian Math. Izv.

40

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
Andrew Pressley and Graeme Segal, Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986. Oxford Science Publications. MR 900587
R. W. Thomason, Homotopy colimits in the category of small categories, Math. Proc. Cambridge Philos. Soc. 85 (1979), no. 1, 91–109. MR 510404, DOI 10.1017/S0305004100055535