Moscow-Petersburg seminar on Low-Dimensional Mathematics July 4, 2003, 16:00--17:45 Pavel Etingof (MIT) Double affine Hecke algebras, Calogero-Moser spaces, and affine cubic surfaces. Abstract: I will speak about the connections of the Double affne Hecke algebras and algebraic geometry. Double affine Hecke algebras H were introduced about 10 years ago by Ivan Cherednik, in order to prove the famous Macdonald conjectures. More specifically, they arose as algebras controlling Macdonald's polynomials (or Ruijsenaars-Schneider integrable systems). These algebras can be degenerated to H' and further to H'' which control Jack polynomials (trigonometric Calogero-Moser systems) and multivariable Bessel functions (rational Calogero-Moser systems). These algebras have many applications in representation theory and integrable systems. Recently V. Ginzburg and I showed that in the classical case (the "Planck constant" equal to zero), in type A, the algebra H'' is the algebra of endomorphisms of an n!-dimensional vector bundle on the Calogero-Moser space of Kazhdan, Kostant, and Sternberg (which is the correct phase space for the classical Calogero-Moser system with rational potential). Thus double affine Hecke algebras define a quantization of the Calogero-Moser space (together with this vector bundle) This result has many applications, including the computation of the cohomology ring of the Calogero-Moser space (and hence the Hilbert scheme of Hilb_n(C^2), which is diffeomorphic to it by Nakajima's theorem). The result was generalized by A. Oblomkov to the cases H' and H. In the second half of the talk I will describe the recent work of A. Oblomkov, which connects double affine Hecke algebras (DAHA) of rank 1 (a 5-parameter family of algebras controlling the most general q-hypergeometric orthogonal polynomials, the Askey-Wilson polynomials) and the algebraic geometry of affine cubic surfaces, obtained by removing a triangle of lines from a projective cubic surface (in the rank 1 case, these surfaces play the role of the Calogero-Moser space). More precisely, such cubic surfaces have a Poisson structure, and double affine Hecke algebras define their quantization. This comparison allows one to interpret geometrically the modular group symmetry of the double affine Hecke algebras, as well as to compute the cohomology of these algebras. In particular, the second Betti number of a generic DAHA is 5; this implies that generic DAHA don't have any deformations other than by variation of the parameters. This explains conceptually the well known fact that Askey-Wilson polynomials are the "most general", i.e. one "cannot" add more parameters to them. --- http://www.pdmi.ras.ru/~lowdimma