Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n)

Abstract

The goal of this paper is to lay the foundations for a combinatorial study, via orthogonal functions and intertwining operators, of category O for the rational Cherednik algebra of type G(r,p,n). As a first application, we give a self-contained and elementary proof of the analog for the groups G(r,p,n), with r>1, of Gordon's theorem (previously Haiman's conjecture) on the diagonal coinvariant ring. We impose no restriction on p; the result for p<r has been proved by Vale using a technique analogous to Gordon's. Because of the combinatorial application to Haiman's conjecture, the paper is logically self-contained except for standard facts about complex reflection groups. The main results should be accessible to mathematicians working in algebraic combinatorics who are unfamiliar with the impressive range of ideas used in Gordon's proof of his theorem.

20 pages; in the 3rd version we have omitted some well-known material to make the paper shorter and included a proof of the analog of Gordon's theorem on the diagonal coinvariant ring for the group G(r,p,n) that avoids the KZ functor

References in corpus (6)

• A recursion and a combinatorial formula for Jack polynomials
• On the quotient ring by diagonal harmonics
• Orthogonal functions generalizing Jack polynomials
• Drinfeld-Hecke algebras over cocommutative algebras
• On category O for the rational Cherednik algebra of G(m,1,n): the almost semisimple case
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Cited by in corpus (8)

• Unitary representations of rational Cherednik algebras
• Orthogonal functions generalizing Jack polynomials
• Category O for Rational Cherednik Algebras H_{t,c}(GL_2(F_p),h) in Characteristic p
• Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth)
• Blocks of restricted rational Cherednik algebras for $G(m,d,n)$
• Jack polynomials and the coinvariant ring of $G(r,p,n)$
• Dirac cohomology and simple modules of the Dunkl-Opdam subalgebra via inherited Drinfeld properties
• q,t-Fuss-Catalan numbers for finite reflection groups