Geometric quantization and families of inner products

Abstract

We formulate a quantization commutes with reduction principle in the setting where the Lie group $G$, the symplectic manifold it acts on, and the orbit space of the action may all be noncompact. It is assumed that the action is proper, and the zero set of a deformation vector field, associated to the momentum map and an equivariant family of inner products on the Lie algebra $\mathfrak{g}$ of $G$, is $G$-cocompact. The central result establishes an asymptotic version of this quantization commutes with reduction principle. Using an equivariant family of inner products on $\mathfrak{g}$ instead of a single one makes it possible to handle both noncompact groups and manifolds, by extending Tian and Zhang's Witten deformation approach to the noncompact case.

72 pages. Minor corrections after comments from a referee

References in corpus (2)

• Applications of Equivariant Cohomology
• The index theory on non-compact manifolds with proper group action

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