Orthogonal functions generalizing Jack polynomials

Abstract:

The rational Cherednik algebra $\mathbb {H}$ is a certain algebra of differential-reflection operators attached to a complex reflection group $W$ and depending on a set of central parameters. Each irreducible representation $S^\lambda$ of $W$ corresponds to a standard module $M(\lambda )$ for $\mathbb {H}$. 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 $\mathfrak {t}$ of $\mathbb {H}$ discovered by Dunkl and Opdam. In this case, the irreducible $W$-modules are indexed by certain sequences $\lambda$ of partitions. We first show that $\mathfrak {t}$ 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 $\mathbb {C}^n$ 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. A consequence of our results is the construction of a number of interesting finite dimensional modules with intricate structure. Finally, we show that for a certain choice of parameters there is a cyclic group of automorphisms of $\mathbb {H}$ so that the rational Cherednik algebra for $G(r,p,n)$ is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for $G(r,p,n)$ by Clifford theory.