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 Z^2, where l is 2,3,4, and 6, respectively. In this
paper, we define a flat deformation H(t,q) of the group algebra 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^\check C_1, which was studied by Noumi, Sahi, and Stokman, and
controls Askey-Wilson polynomials. We prove that H(t,q) is the universal deformation of the
twisted group algebra of G, and that this deformation is compatible with certain
filtrations on C[G]. We also show 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. We also discuss connections of H(t,q) with preprojective algebras and
Painlev
e VI.