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International Mathematics Research Papers (2005) 2005:183-236, doi:10.1155/IMRP.2005.183
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Copyright © 2005 Hindawi Publishing Corporation. All rights reserved.

Graded multiplicities in the Macdonald kernel. Part I

John R. Stembridge

The Macdonald kernel is a virtual character, depending on parameters q and t, for the adjoint form of a complex semisimple Lie group. At Formula, it specializes to the graded characters of the symmetric and exterior algebras of the adjoint representation; in both of these cases, the irreducible decomposition of this character is of significant interest. The Macdonald kernel may also be used to define the bilinear form relative to which the Macdonald polynomials are orthogonal, and the irreducible decomposition of the kernel may be used to compute the Macdonald polynomials explicitly. In this paper, we provide uniform recurrences for computing this irreducible decomposition, prove explicit formulas for the multiplicities of the adjoint and "short adjoint" representations (extending recent work of Bazlov), and provide an explicit formula for the denominators of the graded multiplicities (suitably normalized).


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