Unitary representations of rational Cherednik algebras

Abstract

We study unitarity of lowest weight irreducible representations of rational Cherednik algebras. We prove several general results, and use them to determine which lowest weight representations are unitary in a number of cases. In particular, in type A, we give a complete description of the unitarity locus (justified in Section 5 and appendix by Stephen Griffeth), and resolve a question by Cherednik on the unitarity of the irreducible subrepresentation of the polynomial representation. Also, as a by-product, we establish Kasatani's conjecture in full generality (the previous proof by Enomoto assumes that the parameter c is not a half-integer).

31 pages, latex; the present version contains an appendix by Stephen Griffeth, which allowed to make Conjecture 5.5 from the previous version into Theorem 5.5

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• Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n)
• Orthogonal functions generalizing Jack polynomials
• Non-semisimple Macdonald polynomials
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Cited by in corpus (11)

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• Symmetric and Antisymmetric Vector-valued Jack Polynomials
• Generalized Calogero-Moser systems from rational Cherednik algebras
• Vector-Valued Polynomials and a Matrix Weight Function with $B_2$-Action
• The Smallest Singular Values and Vector-Valued Jack Polynomials
• Orthogonality Measure on the Torus for Vector-Valued Jack Polynomials
• On representations of rational Cherednik algebras in complex rank
• Vector Polynomials and a Matrix Weight Associated to Dihedral Groups
• Unitary representations of Hecke algebras of complex reflection groups
• The smallest singular values of the icosahedral group
• Dunkl Operators and Canonical Invariants of Reflection Groups