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Riemann-Roch-Hirzebruch integral formula for characters of reductive Lie groups

Author: Matvei Libine
Journal: Represent. Theory 9 (2005), 507-524
MSC (2000): Primary 22E45; Secondary 32C38, 19L10, 55N91
Published electronically: August 29, 2005
MathSciNet review: 2167904
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Abstract: 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

Keywords: Equivariant sheaves and ${\mathcal D}$-modules, characteristic cycles of sheaves and ${\mathcal D}$-modules, integral character formula, fixed point integral localization formula, fixed point character formula, representations of reductive Lie groups, equivariant forms
Received by editor(s): January 25, 2004
Received by editor(s) in revised form: February 23, 2005
Published electronically: August 29, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.