A fixed point formula and Harish-Chandra's character formula

Abstract

The main result in this paper is a fixed point formula for equivariant indices of elliptic differential operators, for proper actions by connected semisimple Lie groups on possibly noncompact manifolds, with compact quotients. For compact groups and manifolds, this reduces to the Atiyah-Segal-Singer fixed point formula. Other special cases include an index theorem by Connes and Moscovici for homogeneous spaces, and an earlier index theorem by the second author, both in cases where the group acting is connected and semisimple. As an application of this fixed point formula, we give a new proof of Harish-Chandra's character formula for discrete series representations.

arXiv comment45 pages; corrections and clarifications

References in corpus (6)

• Parabolic induction and restriction via C*-algebras and Hilbert C*-modules
• Quantising proper actions on Spin${}^c$-manifolds
• Positive Scalar Curvature and Poincare Duality for Proper Actions
• Spin-structures and proper group actions
• A fixed point theorem on noncompact manifolds
• An equivariant index for proper actions II: properties and applications

Cited by in corpus (11)

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• A fixed point theorem on noncompact manifolds
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• An index theorem for higher orbital integrals
• Higher genera for proper actions of Lie groups
• Higher genera for proper actions of Lie groups, Part 2: the case of manifolds with boundary
• Higher orbital integrals, rho numbers and index theory
• An equivariant orbifold index for proper actions
• An absolute version of the Gromov-Lawson relative index theorem
• Shelstad's character identity from the point of view of index theory
• Higher localised $\hat{A}$-genera for proper actions and applications