Abstract

We study the algebraic properties of the five-parameter family H(t1,t2,t3,t4;q) of double affine Hecke algebras of type CC1. 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=e provides its deformation quantization. Using that H2(C,ℂ)=ℂ5, we find that the Hochschild cohomology HH2(H) (for q=e) 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¯t¯4, gives the universal deformation of the Poisson algebra [X±1,P±1]2.

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