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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The Signature of the Shapovalov Form on Irreducible Verma Modules
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by Wai Ling Yee
Represent. Theory 9 (2005), 638-677
Published electronically: December 8, 2005


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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Bibliographic Information
  • Wai Ling Yee
  • Affiliation: Department of Mathematical Sciences, University of Alberta, Edmonton, Alberta, Canada
  • Email:
  • 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:
  • MathSciNet review: 2183058