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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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Riemann-Roch-Hirzebruch integral formula for characters of reductive Lie groups
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by Matvei Libine PDF
Represent. Theory 9 (2005), 507-524 Request permission


Let $G_{\mathbb R}$ be a real reductive Lie group acting on a manifold $M$. M. Kashiwara and W. Schmid constructed representations of $G_{\mathbb R}$ using sheaves and quasi-$G_{\mathbb R}$-equivariant ${\mathcal D}$-modules on $M$. In this article we prove an integral character formula for these representations (Theorem 1). Our main tools will be the integral localization formula recently proved by the author and the integral character formula proved by W. Schmid and K. Vilonen (originally established by W. Rossmann) in the important special case when the manifold $M$ is the flag variety of $\mathbb C \otimes _{\mathbb R} \mathfrak {g}_{\mathbb R}$—the complexified Lie algebra of $G_{\mathbb R}$. In the special case when $G_{\mathbb R}$ is commutative and the ${\mathcal D}$-module is the sheaf of sections of a $G_{\mathbb R}$-equivariant line bundle over $M$ this integral character formula will reduce to the classical Riemann-Roch-Hirzebruch formula. As an illustration we give a concrete example on the enhanced flag variety.
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Additional Information
  • Matvei Libine
  • Affiliation: Department of Mathematics and Statistics, University of Massachusetts, Lederle Graduate Research Tower, 710 North Pleasant Street, Amherst, Massachusetts 01003
  • Address at time of publication: Department of Mathematics, Yale University, P.O. Box 208283, New Haven, Connecticut 06520-8283
  • Email:
  • Received by editor(s): January 25, 2004
  • Received by editor(s) in revised form: February 23, 2005
  • Published electronically: August 29, 2005
  • © Copyright 2005 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Represent. Theory 9 (2005), 507-524
  • MSC (2000): Primary 22E45; Secondary 32C38, 19L10, 55N91
  • DOI:
  • MathSciNet review: 2167904