Deformation theory of the Double Affine Hecke algebra of type $(C_n^\vee,C_n)$

Abstract

We study the double affine Hecke algebra (DAHA) of type $(C_n^\vee,C_n)$ from the perspective of deformation theory. First, we provide a zeros-and-residues realization of this algebra, extending the construction of Ginzburg, Kapranov, and Vasserot to the non-reduced affine root system setting. Specializing the parameters of the DAHA to the base point gives the crossed product of a quantum torus algebra with the finite Weyl group of type $C_n$. We then show that for all $n$, the completed DAHA is the formal universal deformation of this crossed product algebra, extending Oblomkov's result for $n=1$. Our proof explicitly identifies the completed DAHA with the undeformed crossed product algebra equipped with a formal star product.