Double affine Hecke algebras and 2-dimensional local fields

Kapranov, M.¹

¹ Department of Mathematics, University of Toronto, 100 St. George St., Toronto, Ontario, Canada M5S 3G3

Journal of the American Mathematical Society, Tome 14 (2001) no. 1, pp. 239-262

Résumé

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.

Détail

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

Bibliographie

[1] Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos 1972

[2] Artin, M., Mazur, B. Etale homotopy 1986

[3] Brown, Kenneth S. Buildings 1989

[4] Casselman, W. The unramified principal series of 𝔭-adic groups. I. The spherical function Compositio Math. 1980 387 406

[5] Cherednik, Ivan Double affine Hecke algebras and Macdonald’s conjectures Ann. of Math. (2) 1995 191 216

[6] Chriss, Neil, Khuri-Makdisi, Kamal On the Iwahori-Hecke algebra of a 𝑝-adic group Internat. Math. Res. Notices 1998 85 100

[7] Drinfel′D, V. G. 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) 1984 138 156

[8] Gabriel, P., Zisman, M. Calculus of fractions and homotopy theory 1967

[9] Gel′Fand, I. M., Graev, M. I., Pyatetskii-Shapiro, I. I. Representation theory and automorphic functions 1969

[10] Ginzburg, Victor, Kapranov, Mikhail, Vasserot, Eric Residue construction of Hecke algebras Adv. Math. 1997 1 19

[11] Gray, John W. Formal category theory: adjointness for 2-categories 1974

[12] Retakh, V. S. Massey operations in Lie superalgebras and differentials of the Quillen spectral sequence Colloq. Math. 1985 81 94

[13] Kazhdan, David, Lusztig, George Proof of the Deligne-Langlands conjecture for Hecke algebras Invent. Math. 1987 153 215

[14] Lusztig, George Singularities, character formulas, and a 𝑞-analog of weight multiplicities 1983 208 229

[15] Lusztig, George Intersection cohomology methods in representation theory 1990

[16] Matsumoto, Hideya Sur les sous-groupes arithmétiques des groupes semi-simples déployés Ann. Sci. École Norm. Sup. (4) 1969 1 62

[17] Milnor, John Introduction to algebraic 𝐾-theory 1971

[18] Parshin, A. N. Vector bundles and arithmetic groups. I Trudy Mat. Inst. Steklov. 1995 240 265

[19] Pressley, Andrew, Segal, Graeme Loop groups 1986

[20] Thomason, R. W. Homotopy colimits in the category of small categories Math. Proc. Cambridge Philos. Soc. 1979 91 109