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

DOI:
https://doi.org/10.1090/S0002-9947-2010-05156-6

Published electronically:
June 21, 2010

MathSciNet review:
2661511

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Abstract: The rational Cherednik algebra is a certain algebra of differential-reflection operators attached to a complex reflection group and depending on a set of central parameters. Each irreducible representation of corresponds to a standard module for . This paper deals with the infinite family of complex reflection groups; our goal is to study the standard modules using a commutative subalgebra of discovered by Dunkl and Opdam. In this case, the irreducible -modules are indexed by certain sequences of partitions. We first show that acts in an upper triangular fashion on each standard module , with eigenvalues determined by the combinatorics of the set of standard tableaux on . As a consequence, we construct a basis for consisting of orthogonal functions on with values in the representation . For with 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 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 so that the rational Cherednik algebra for is the fixed subalgebra. Our results therefore descend to the rational Cherednik algebra for 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:
https://doi.org/10.1090/S0002-9947-2010-05156-6

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.