Double affine Hecke algebras of rank 1 and affine cubic surfaces

Abstract

We study the algebraic properties of the five-parameter family H(t1,t2,t3,t4;q) of double affine Hecke algebras of type C∨C1⁠. This family generalizes Cherednik's double affine Hecke algebras of rank 1. It was introduced by Sahi and studied by Noumi and Stokman as an algebraic structure which controls Askey-Wilson polynomials. We show that if q=1, then the spectrum of the center of H is an affine cubic surface C, obtained from a projective one by removing a triangle consisting of smooth points. Moreover, any such surface is obtained as the spectrum of the center of H for some values of parameters. This result allows one to give a simple geometric description of the action of an extension of PGL2(ℤ) by ℤ on the center of H. When C is smooth, it admits a unique algebraic symplectic structure, and the spherical subalgebra eHe of the algebra H for q=eℏ provides its deformation quantization. Using that H2(C,ℂ)=ℂ5, we find that the Hochschild cohomology HH2(H) (for q=eℏ⁠) is five-dimensional for generic parameter values. From this we deduce that the only deformations of H come from variations of parameters. This explains from the point of view of noncommutative geometry why one cannot add more parameters into the theory of Askey-Wilson polynomials. We also prove that the five-parameter family H(t1,t2,t3,t4;q) of algebras yields the universal deformation of the semidirect product q-Weyl algebra with Z2 and that the family of cubic surfaces C=Ct¯⁠, t¯∈⅜t¯4⁠, gives the universal deformation of the Poisson algebra ⅜[X±1,P±1]ℤ2⁠.