Representations of rational Cherednik algebras with minimal support and torus knots

Abstract

In this paper we obtain several results about representations of rational Cherednik algebras, and discuss their applications. Our first result is the Cohen–Macaulayness property (as modules over the polynomial ring) of Cherednik algebra modules with minimal support. Our second result is an explicit formula for the character of an irreducible minimal support module in type $A_{n-1}$ for $c=\frac{m}{n}$, and an expression of its quasispherical part (i.e., the isotypic part of “hooks”) in terms of the HOMFLY polynomial of a torus knot colored by a Young diagram. We use this formula and the work of Calaque, Enriquez and Etingof to give explicit formulas for the characters of the irreducible equivariant D-modules on the nilpotent cone for ${\mathit{SL}}_m$. Our third result is the construction of the Koszul–BGG complex for the rational Cherednik algebra, which generalizes the construction of the Koszul–BGG resolution from [3] and [21], and the calculation of its homology in type A. We also show in type A that the differentials in the Koszul–BGG complex are uniquely determined by the condition that they are nonzero homomorphisms of modules over the Cherednik algebra. Finally, our fourth result is the symmetry theorem, which identifies the quasispherical components in the representations with minimal support over the rational Cherednik algebras $H_{\frac{m}{n}}(S_n)$ and $H_{\frac{n}{m}}(S_m)$. In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry was essentially established in [8] in the spherical case, and in [24] in the case $\mathit{GCD}(m,n)=1$, and it has a natural interpretation in terms of invariants of torus knots.