  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
$\mathcal{O}_{G/P}(R)$.
