Polarized orbifolds associated to quantized Hamiltonian torus actions

TL;DR

This paper explores the geometric structure of isotypical components in Hardy spaces associated with Hamiltonian torus actions on polarized manifolds, revealing their relation to polarized orbifolds and generalizing weighted projective space constructions.

Contribution

It provides a geometric interpretation of isotypical components for weights tending to infinity, using polarized orbifolds derived from Hamiltonian actions, extending classical quotient constructions.

Findings

Finite-dimensionality of isotypical components under certain conditions

Geometric interpretation of these components via polarized orbifolds

Generalization of weighted projective space quotients

Abstract

Suppose given an holomorphic and Hamiltonian action of a compact torus $T$ on a polarized Hodge manifold $M$. Assume that the action lifts to the quantizing line bundle, so that there is an induced unitary representation of $T$ on the associated Hardy space. If in addition the moment map is nowhere zero, for each weight $\boldsymbol{\nu}$ the $\boldsymbol{\nu}$-th isotypical component in the Hardy space of the polarization is finite-dimensional. Assuming that the moment map is transverse to the ray through $\boldsymbol{\nu}$, we give a gometric interpretation of the isotypical components associated to the weights $k\,\boldsymbol{\nu}$, $k\rightarrow +\infty$, in terms of certain polarized orbifolds associated to the Hamiltonian action and the weight. These orbifolds are generally not reductions of $M$ in the usual sense, but arise rather as quotients of certain loci in the unit circle bundle of the polarization; this construction generalizes the one of weighted projective spaces as quotients of the unit sphere, viewed as the domain of the Hopf map.