In \cite{ Broer1993}, it was shown that certain line bundles on $\widetilde{ \mathcal{ N}}=T^*G/B$ 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}.
Categories: math.RT math.AG