Pavel Etingof

Generalized double affine Hecke algebras and quantized del Pezzo surfaces

Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D4,E6,E7,E8). To such a diagram one can attach a group $G$ whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: $G=Z_l\ltimes Z2$, where $l$ is 2,3,4, and 6, respectively. I will define a flat deformation $H(t,q)$ of the group algebra $\bold C[G]$ of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra $H(t,q)$ for D4 is the Cherednik algebra of type $C^\vee C_1$, which was studied by Noumi, Sahi, and Stokman, and controls Askey-Wilson polynomials. I'll explain that the algebra $H(t,q)$ is the universal deformation of the twisted group algebra of $G$, and this deformation is compatible with certain filtrations on $\Bbb C[G]$. I will also explain that if $q$ is a root of unity, then for generic $t$ the algebra $H(t,q)$ is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra $eH(t,q)e$ provides a quantization of such surfaces. Finally, I'll discuss connections of H(t,q) with preprojective algebras and equation "Painlev\'e VI". This is joint work with Alex Oblomkov and Eric Rains.