Representations of reductive groups in cohomology spaces
Consider a connected complex reductive group \(G\) acting on a complex algebraic variety \(X\). Recall that for any \(G\)-linearized coherent sheaf \({\mathcal F}\) on \(X\), the cohomology spaces \(H^ i (X, {\mathcal F})\) are rational \(G\)-modules: they decompose into a direct sum of simple \(G\)- modules, with (finite or infinite) multiplicities. We prove the following alternative: (1) If a Borel subgroup of \(G\) has a dense orbit in \(X\), then there is a (finite) upper bound \(C(X, {\mathcal F})\) for all multiplicities of the \(G\)-modules \(H^ i (X, {\mathcal F})\). In fact, there exists a constant \(C(X)\) such that for any \(G\)-linearized vector bundle \(E\) over \(X\), and for any integer \(i \geq 0\), the multiplicities of \(H^ i (X,E)\) are at most \(C(X) \cdot\text{rk} (E)\). (2) If no Borel subgroup has a dense orbit on \(X\), then there exists a \(G\)-variety \(Y\) which is \(G\)-birational to \(X\), and a \(G\)-linearized invertible sheaf \(L\) over \(X\), such that the subspace of \(G\)-invariants in \(H^ 0 (X,L)\) or \(H^ 1 (X,L)\) is infinite-dimensional. The first statement is a qualitative analog of the classical Borel-Weil-Bott theorem, which describes the cohomology groups of line bundles on flag varieties.
cohomology groups of line bundles on flag varieties
scientific article; zbMATH DE number 734588