Representation theory of the global Cherednik algebra

Abstract

This thesis studies the representation theory of Cherednik algebras associated to a complex algebraic varieties which carries the action of a finite group. First, we prove that the Knizhnik-Zamolodchikov functor from the category of P-coherent modules for the Cherednik algebra to finite dimensional modules over the Hecke algebra is essentially surjective. Then we begin to use this result to study the analog of category 0 for Cherednik algebras on Riemann surfaces and on products of elliptic curves. In particular we give conditions on the parameters under which this category 0 analog is nonzero. Our next goal is to generalize several basic results from the theory of D-modules to the representation theory of rational Cherednik algebras. We relate characterizations of holonomic modules in terms of singular support and Gelfand-Kirillov dimension. We study pullback, pushforward, and dual on the derived category of (holonomic) Cherednik modules for certain classes of maps between varieties. We prove, in the case of generic parameters for the rational Cherednik algebra, that pushforward with respect to an open affine inclusion preserves holonomicity. Finally, we relate the global sections algebra of the sheaf of Cherednik algebras associated to a smooth quadric hypersurface of Pn to the Dunkl angular momentum algebra of Feigin-Hakobyan. In particular, this lets us relate the angular momentum algebra for a rank 3 Coxeter group to the rank 2 symplectic reflection algebra for a corresponding finite subgroup of SL2.