DMCA
Orthogonal functions generalizing Jack polynomials
| Venue: | Trans. Amer. Math. Soc |
| Citations: | 16 - 9 self |
BibTeX
@ARTICLE{Griffeth_orthogonalfunctions,
author = {Stephen Griffeth},
title = {Orthogonal functions generalizing Jack polynomials},
journal = {Trans. Amer. Math. Soc},
year = {}
}
Share
 |
 |
 |
 |
||
OpenURL
Abstract
Abstract. The rational Cherednik algebra 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 λ of W corresponds to a standard module M(λ) for 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 t of H discovered by Dunkl and Opdam. In this case, the irreducible W-modules are indexed by certain sequences λ of partitions. We first show that t acts in an upper triangular fashion on each standard module M(λ), with eigenvalues determined by the combinatorics of the set of standard tableaux on λ. As a consequence, we construct a basis for M(λ) consisting of orthogonal functions on C n with values in the representation S λ. For G(1, 1, n) with λ = (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(λ) 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 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.
Keyphrases
orthogonal function rational cherednik algebra jack polynomial standard module complex reflection group fixed subalgebra central parameter differential-reflection operator non-symmetric jack polynomial irreducible w-modules certain sequence explicit combinatorial description certain algebra intricate structure clifford theory norm formula commutative subalgebra irreducible representation cyclic group upper triangular fashion certain choice standard tableau finite dimensional module infinite family
