Representations of Rational Cherednik algebras with minimal support and torus knots

TL;DR

This paper explores the structure and representations of rational Cherednik algebras with minimal support, revealing their Cohen-Macaulay properties, explicit character formulas linked to torus knots, a generalized Koszul-BGG complex, and a symmetry relating different algebraic representations.

Contribution

It introduces new results on minimal support modules, explicit character formulas connected to HOMFLY polynomials, a generalized Koszul-BGG complex, and a symmetry theorem in rational Cherednik algebra representations.

Findings

Modules with minimal support are Cohen-Macaulay.

Explicit character formulas relate to torus knot invariants.

A generalized Koszul-BGG complex is constructed and analyzed.

Abstract

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=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 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 by Berest-Etingof-Ginzburg and Gordon, 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_{m/n}(S_n) and H_{n/m}(S_m). In fact, we show that the simple quotients of the corresponding quasispherical subalgebras are isomorphic as filtered algebras. This symmetry has a natural interpretation in terms of invariants of torus knots.