These are the notes for a short course given at the summer school Equivariant Combinatorics at the CRM in Montreal. The notes contain somewhat more material than was practical to cover in the course. The intended audience was graduate students and researchers in algebraic combinatorics with no prior experience with Cherednik algebras, but who are interested in the algebraic combinatorics having...

In the first part of this article we review the general theory of Cherednik algebras and non-symmetric Macdonald polynomials, including a formulation and proof of the fundamental duality theorem in its proper general context. In the last section we summarize some of the combinatorial results in this area which we have recently obtained in collaboration with J. Haglund and N. Loehr. Mathematics ...

We determine the structure of category O for the rational Cherednik algebra of G(m, 1, n) in the case where the KZ functor satisfies a condition called separating simples. We show that this condition holds whenever all but one of the simple objects in O has the maximum possible Gelfand-Kirillov dimension (plus a mild genericity condition on the parameters). Our proof involves calculating the bl...

Nonsymmetric Koornwinder polynomials are multivariable extensions of nonsymmetric Askey-Wilson polynomials. They naturally arise in the representation theory of (double) affine Hecke algebras. In this paper we discuss how nonsymmetric Koornwinder polynomials naturally arise in the theory of the Heisenberg XXZ spin1 2 chain with general reflecting boundary conditions. A central role in this stor...

Abstract We provide a homological construction of unitary simple modules Cherednik and Hecke algebras type A via BGG resolutions, solving conjecture Berkesch–Griffeth–Sam. vastly generalize the its solution to cyclotomic over arbitrary ground fields, calculate Betti numbers Castelnuovo–Mumford regularity certain symmetric linear subspace arrangements.

In this note, given an associative algebra A over C, we compute the Hochschild homology of the crossed product C[Sn]⋉A . If A is simple, this homology coincides with the homology of SA. If A satisfies the “Gorenstein” properties of [VB1, VB2], then this computation allows us to compute the Hochschild cohomology of C[Sn] ⋉ A , and of SA for simple A. In particular, we obtain a result conjectured...