K-invariant Hilbert Modules and Singular Vector Bundles on Bounded Symmetric Domains

TL;DR

This paper demonstrates that eigenbundles of specific Hilbert modules over bounded symmetric domains are singular vector bundles that decompose into stratified sums of homogeneous bundles, with fibers described via representation theory.

Contribution

It introduces the concept of singular vector bundles arising from Hilbert modules on symmetric domains and details their stratified decomposition and fiber realization.

Findings

Eigenbundles are singular vector bundles with stratified structure.

Decomposition into homogeneous vector bundles along a canonical stratification.

Fibers characterized through representation theory on normal spaces.

Abstract

We show that the "eigenbundle" (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank r is a "singular" vector bundle (linearly fibrered complex analytic space) which decomposes as a stratified sum of homogeneous vector bundles along a canonical stratification of length r+1. The fibres are realized in terms of representation theory on the normal space of the strata.