Symplectic Induction, Prequantum Induction, and Prequantum Multiplicities

TL;DR

This paper explores the relationship between symplectic induction, prequantum induction, and multiplicities, showing that working within prequantum G-spaces resolves ambiguities present in Hamiltonian G-spaces and restores Frobenius reciprocity.

Contribution

It introduces a framework of prequantum G-spaces to establish Frobenius reciprocity and induction properties that fail in Hamiltonian G-spaces.

Findings

Prequantum induction restores Frobenius reciprocity.

Ambiguity in quantization of Hamiltonian G-spaces is resolved.

Induction in stages property is established in prequantum setting.

Abstract

Frobenius reciprocity asserts that induction from a subgroup and restriction to it are adjoint functors in categories of unitary G-modules. In the 1980s, Guillemin and Sternberg established a parallel property of Hamiltonian G-spaces, which (as we show) unfortunately fails to mirror the situation where more than one G-module "quantizes" a given Hamiltonian G-space. This paper offers evidence that the situation is remedied by working in the category of *prequantum* G-spaces, where this ambiguity disappears; there, we define induction and multiplicity spaces, and establish Frobenius reciprocity as well as the "induction in stages" property.