Vanishing Cohomology of Dominant Line Bundles for Real Groups

Abstract

In \cite{Broer1993}, it was shown that certain line bundles on $\tilde{\mathcal{N}}=T^{\ast }G/B$ have vanishing higher cohomology. We prove a generalization of this theorem for real reductive algebraic groups. More specifically, if ${\mathcal{N}}_{\theta }$ denotes the cone of nilpotent elements in a Cartan subspace $\mathfrak{p},$ we have a similar construction of a resolution of singularities $\widetilde{{\mathcal{N}}_{\theta }}.$ We prove that for a certain cone of weights $H^i(\widetilde{{\mathcal{N}}_{\theta }},{\mathcal{O}}_{\widetilde{{\mathcal{N}}_{\theta }}}(\lambda ))=0$ for $i>0.$ This follows by combining a simple calculation of the canonical bundle for $\widetilde{{\mathcal{N}}_{\theta }}$ with Grauert-Riemenschneider vanishing. Restricting to the structure sheaf, we get a characterization of the singularities of the normalization of ${\mathcal{N}}_{\theta }.$ We use this to show that for groups of QCT (Definition 2), $\mathbb{C}[{\mathcal{N}}_{\theta }]$ is equivalent as a $K$-representation to a certain cohomologically induced module giving a new proof of a result in \cite{KostantRallis1971}.

13 pages. Comments welcomed