Orthogonal functions generalizing Jack polynomials

Abstract

The rational Cherednik algebra $\HH$ is a certain algebra of differential-reflection operators attached to a complex reflection group $W$. Each irreducible representation $S^{\lambda }$ of $W$ corresponds to a standard module $M(\lambda )$ for $\HH$. This paper deals with the infinite family $G(r,1,n)$ of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra $\ttt$ of $\HH$ discovered by Dunkl and Opdam. In this case, the irreducible $W$-modules are indexed by certain sequences $\lambda$ of partitions. We first show that $\ttt$ acts in an upper triangular fashion on each standard module $M(\lambda )$, with eigenvalues determined by the combinatorics of the set of standard tableaux on $\lambda$. As a consequence, we construct a basis for $M(\lambda )$ consisting of orthogonal functions on $\CC^n$ with values in the representation $S^{\lambda }$. For $G(1,1,n)$ with $\lambda =(n)$ these functions are the non-symmetric Jack polynomials. We use intertwining operators to deduce a norm formula for our orthogonal functions and give an explicit combinatorial description of the lattice of submodules of $M(\lambda )$ in the case in which the orthogonal functions are all well-defined.

21 pages; revised version contains a combinatorial description of the set of submodules of each standard module; 2nd revision uses Clifford theory to relate G(r,p,n) Cherednik algebra to that for G(r,1,n)

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Cited by in corpus (7)

• Unitary representations of rational Cherednik algebras
• Towards a combinatorial representation theory for the rational Cherednik algebra of type G(r,p,n)
• Symmetric and Antisymmetric Vector-valued Jack Polynomials
• Generalized Jack polynomials and the representation theory of rational Cherednik algebras
• The Smallest Singular Values and Vector-Valued Jack Polynomials
• A Linear System of Differential Equations Related to Vector-Valued Jack Polynomials on the Torus
• Jack polynomials and the coinvariant ring of $G(r,p,n)$