Mathematics – Representation Theory
Scientific paper
2007-07-02
Mathematics
Representation Theory
21 pages; revised version contains a combinatorial description of the set of submodules of each standard module; 2nd revision
Scientific paper
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.
No associations
LandOfFree
Orthogonal functions generalizing Jack polynomials does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Orthogonal functions generalizing Jack polynomials, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Orthogonal functions generalizing Jack polynomials will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-692898