Decomposition of cohomology of vector bundles on homogeneous ind-spaces

TL;DR

This paper investigates how the cohomology of vector bundles on homogeneous ind-spaces decomposes into simpler parts, extending classical finite-dimensional results to an infinite-dimensional setting without relying on semisimplicity.

Contribution

It establishes a decomposition of cohomology for vector bundles on homogeneous ind-spaces under certain conditions, generalizing Bott-Borel-Weil to infinite-dimensional ind-groups.

Findings

Cohomology decomposes into sums of simpler bundle cohomologies.

Results extend classical finite-dimensional theorems to infinite-dimensional cases.

Injectivity of cohomologies is key in the infinite-dimensional analysis.

Abstract

Let $G$ be a locally semisimple ind-group, $P$ be a parabolic subgroup, and $E$ be a finite-dimensional $P$-module. We show that, under a certain condition on $E$, the nonzero cohomologies of the homogeneous vector bundle $\mathcal{O}_{G/P}(E^*)$ on $G/P$ induced by the dual $P$-module $E^*$ decompose as direct sums of cohomologies of bundles of the form $\mathcal{O}_{G/P}(R)$ for (some) simple constituents $R$ of $E^*$. In the finite-dimensional case, this result is a consequence of the Bott-Borel-Weil theorem and Weyl's semisimplicity theorem. In the infinite-dimensional setting we consider, there is no relevant semisimplicity theorem. Instead, our results are based on the injectivity of the cohomologies of the bundles $\mathcal{O}_{G/P}(R)$.