Double Affine Hecke Algebras and 2-dimensional Local Fields

The concept of an n-dimensional local field was introduced by A.N. Parshin [Pa1] with the aim of generalizing the classical adelic formalism to (absolutely) ndimensional schemes. By definition, a 0-dimensional local field is just a finite field, and an n-dimensional local field, n > 0, is a complete discrete valued field whose residue field is (n − 1)-dimensional local. Thus for n = 1 we get locally compact fields such as Qp, Fq((t)), and for n = 2 we get fields such as Qp((t)), Fq((t1))((t2)), etc. In representation theory, harmonic analysis on reductive groups over 0and 1dimensional local fields leads, in particular, to consideration of the finite and affine Hecke algebras Hq, H • q associated to any finite root system R and any q ∈ C∗. These algebras can be defined in several ways, one being by generators and relations, another as the convolution algebra, with respect to the Haar measure, of functions on the group bi-invariant with respect to an appropriate subgroup (i.e., as the algebra of double cosets). Harmonic analysis on groups over 2-dimensional local fields has not been developed, the main difficulty being the infinite dimensionality (absense of local compactness) of such fields. However, the double affine Hecke algebra H •• q recently defined by I. Cherednik [Ch] in terms of generators and relations, looks like the third term in the hierarchy starting from Hq, H • q. The problem “give a group-theoretic construction of the Cherednik algebra” (i.e., realize it as some algebra of double cosets) was proposed by D. Kazhdan a few years ago. The purpose of the present paper is to provide a solution to this problem by developing beginnings of harmonic analysis on reductive groups over 2-dimensional local fields. We consider a simple algebraic group G (over Z), a 2-dimensional local field K = k((t)) of equal characteristic (so k is 1-dimensional local) and the canonical central extension Γ of G(K) by k∗. For an appropriate subgroup ∆1 ⊂ Γ the fibers of the Hecke correspondences (3.1) are locally compact spaces (affine spaces over k of growing dimension) which possess natural invariant measures, so one can formally define the Hecke operators associated to double cosets by integrating over these measures. The main difficulty here is the noncompactness of the domain of integration. It poses convergence problems, making it unclear how to compose such operators or how to define their action from some vector space to itself. More precisely, the operators are well defined on the space F0 of functions on Γ/∆1 with certain proper support conditions but their values lie in a bigger space F .