Symmetric and exterior powers of homogeneous vector bundles (Q1337554)

The aim of this paper is to prove the following theorem, which is a generalization of a question due to Y. Laszlo: Let \(G\) be a connected semi-simple algebraic group over an algebraically closed field \(K\) of \(\text{char }0\), and let \(P\) be a parabolic subgroup. Let \(V_ 1\) and \(V_ 2\) be two completely reducible \(P\)-modules, such that both of them are \(P\)-submodules of certain \(G\)-modules. Then the canonical map, induced from the diagonal embedding \(G/P \hookrightarrow G/P \times G/P\), \[ \beta : H^ 0(G/P \times G/P, {\mathcal L}(V^*_ 1) \boxtimes {\mathcal L}(V^*_ 2)) \to H^ 0(G/P, {\mathcal L}(V^*_ 1 \otimes V^*_ 2)) \] is surjective; where \({\mathcal L}(V^*)\) denotes the vector bundle on the base space \(G/P\) associated to the principal \(P\)-bundle \(G \to G/P\) via the representation \(V^*\) of \(P\), and \({\mathcal L} (V^*_ 1) \boxtimes {\mathcal L}(V^*_ 2)\) denotes the external tensor product of the vector bundles \({\mathcal L}(V^*_ 1)\) and \({\mathcal L}(V^*_ 2)\).