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.