Stratified Hilbert Modules on Bounded Symmetric Domains

TL;DR

This paper studies the structure of Hilbert modules over bounded symmetric domains, revealing a stratified geometric framework with fibers linked to flag manifolds and reproducing kernel derivatives.

Contribution

It introduces a novel stratification of eigenbundles in Hilbert modules on symmetric domains, connecting complex geometry, kernel functions, and determinantal ideals.

Findings

Fibers described as line bundle sections over flag manifolds

Metric embedding via derivatives of reproducing kernels

Stratification of eigenbundles of length r+1

Abstract

We analyze the "eigenbundle" (localization bundle) of certain Hilbert modules over bounded symmetric domains of rank $r,$ giving rise to complex-analytic fibre spaces which are stratified of length $r+1.$ The fibres are described in terms of K\"ahler geometry as line bundle sections over flag manifolds, and the metric embedding is determined by taking derivatives of reproducing kernel functions. Important examples are the determinantal ideals defined by vanishing conditions along the various strata of the stratification.