Decomposition of cohomology of vector bundles on homogeneous ind-spaces

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 ${\mathcal{O}}_{G/P}(E^{\ast })$ on $G/P$ induced by the dual $P$-module $E^{\ast }$ decompose as direct sums of cohomologies of bundles of the form ${\mathcal{O}}_{G/P}(R)$ for (some) simple constituents $R$ of $E^{\ast }$. 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 ${\mathcal{O}}_{G/P}(R)$.

arXiv comment8 pages

References in corpus (1)

• Branching laws for tensor modules over classical locally finite Lie algebras