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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Orthogonal functions generalizing Jack polynomials


Author: Stephen Griffeth
Journal: Trans. Amer. Math. Soc. 362 (2010), 6131-6157
MSC (2010): Primary 05E05, 05E10, 05E15, 16S35, 20C30; Secondary 16D90, 16S38, 16T30
Published electronically: June 21, 2010
MathSciNet review: 2661511
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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.


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Additional Information

Stephen Griffeth
Affiliation: School of Mathematics, University of Minnesota, 127 Church Street, Minneapolis, Minnesota 55455
Address at time of publication: School of Mathematics, James Clerk Maxwell Building, University of Edinburgh, Edinburgh, EH9 3JZ, United Kingeom
Email: griffeth@math.umn.edu, S.Griffeth@ed.ac.uk

DOI: http://dx.doi.org/10.1090/S0002-9947-2010-05156-6
PII: S 0002-9947(2010)05156-6
Received by editor(s): November 20, 2008
Received by editor(s) in revised form: July 3, 2009
Published electronically: June 21, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.