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The Signature of the Shapovalov Form on Irreducible Verma Modules

Author: Wai Ling Yee
Journal: Represent. Theory 9 (2005), 638-677
MSC (2000): Primary 22E47; Secondary 20F55
Published electronically: December 8, 2005
MathSciNet review: 2183058
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Abstract: A Verma module may admit an invariant Hermitian form, which is unique up to a real scalar when it exists. Suitably normalized, it is known as the Shapovalov form. The collection of highest weights decomposes under the affine Weyl group action into alcoves. The signature of the Shapovalov form for an irreducible Verma module depends only on the alcove in which the highest weight lies. We develop a formula for this signature, depending on the combinatorial structure of the affine Weyl group.

Classifying the irreducible unitary representations of a real reductive group is equivalent to the algebraic problem of classifying the Harish-Chandra modules admitting a positive definite invariant Hermitian form. Finding a formula for the signature of the Shapovalov form is a related problem which may be a necessary first step in such a classification.

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Additional Information

Wai Ling Yee
Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada

Received by editor(s): January 18, 2005
Published electronically: December 8, 2005
Additional Notes: This research was supported in part by an NSERC postgraduate fellowship and by an NSF research assistantship.
Article copyright: © Copyright 2005 American Mathematical Society