Double affine Hecke algebras and 2-dimensional local fields

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 G by a subgroup F \mathcal F , extending the well-known interpretations of the finite and affine Hecke algebras. In this interpretation, G G consists of K K -points of a simple algebraic group, where K K is a 2-dimensional local field such as Q p ( ( t ) ) \mathbf Q_p((t)) or F q ( ( t 1 ) ) ( ( t 2 ) ) F_q((t_1))((t_2)) , and F \mathcal F is a certain analog of the Iwahori subgroup.