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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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The Signature of the Shapovalov Form on Irreducible Verma Modules
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by Wai Ling Yee PDF
Represent. Theory 9 (2005), 638-677 Request permission

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
  • Email: wlyee@math.ualberta.ca
  • 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.
  • © Copyright 2005 American Mathematical Society
  • Journal: Represent. Theory 9 (2005), 638-677
  • MSC (2000): Primary 22E47; Secondary 20F55
  • DOI: https://doi.org/10.1090/S1088-4165-05-00269-4
  • MathSciNet review: 2183058