  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.
