Vanishing Cohomology of Dominant Line Bundles for Real Groups

TL;DR

The paper extends vanishing results for higher cohomology of line bundles from complex to real reductive groups by constructing a $K$-equivariant Springer-type resolution of the real nilpotent cone $\mathcal{N}_\theta$ and computing its canonical bundle. Using a Grauert–Riemenschneider vanishing argument, it proves $H^i(\widetilde{\mathcal{N}_\theta}', \mathcal{O}(\lambda'))=0$ for $i>0$ with $\lambda'$ in the appropriate Weyl monoid, and deduces rational singularities for the normalization of $\mathcal{N}_\theta$ along with a realization of $\mathbb{C}[\mathcal{N}_\theta]$ as a cohomologically induced module in the quasi-complex-type (QCT) setting. The work further shows that the normalization and $K$-orbit components have rational singularities, and provides a new proof of Kostant–Rallis type results via these $K$-equivariant constructions. It also outlines a path to broader generalizations under QCT and QAT hypotheses, including complete-intersection and global-function characterizations of $\mathcal{N}_\theta$.

Abstract

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 $\mathcal{N}_θ$ denotes the cone of nilpotent elements in a Cartan subspace $\mathfrak{p},$ we have a similar construction of a resolution of singularities $\widetilde{\mathcal{N}_θ}.$ We prove that for a certain cone of weights $H^i(\widetilde{\mathcal{N}_θ},\mathcal{O}_{\widetilde{\mathcal{N}_θ}}(λ))=0$ for $i> 0.$ This follows by combining a simple calculation of the canonical bundle for $\widetilde{\mathcal{N}_θ}$ with Grauert-Riemenschneider vanishing. Restricting to the structure sheaf, we get a characterization of the singularities of the normalization of $\mathcal{N}_θ.$ We use this to show that for groups of QCT (Definition 2), $\mathbb{C}[\mathcal{N}_θ]$ is equivalent as a $K$-representation to a certain cohomologically induced module giving a new proof of a result in \cite{KostantRallis1971}.

Theorems & Definitions (49)

• Theorem
• Lemma
• Definition 1
• Theorem 1.1: Corollary \ref{['Nilpotent_Orbit_Case']}
• Theorem 1.2: Theorem \ref{['Vanishing_of_Higher_Cohomology']}
• Theorem 1.3: Corollary \ref{['Rational_normal']}
• Lemma 2.1
• proof
• Lemma 2.2: Kostant Kostant1963
• proof