On equivariant principal bundles over wonderful compactifications

TL;DR

This paper proves polystability of certain equivariant principal bundles and stability of tangent bundles over wonderful compactifications of symmetric spaces, extending understanding of geometric structures on these spaces.

Contribution

It establishes conditions under which equivariant principal bundles are polystable and shows tangent bundle stability for specific wonderful compactifications.

Findings

Equivariant principal bundles with irreducible isotropy action are polystable.

Tangent bundles of certain wonderful compactifications are stable.

Results apply to compactifications of quotients of PSL(n,C) by specific subgroups.

Abstract

Let $G$ be a simple algebraic group of adjoint type over $\mathbb C$, and let $M$ be the wonderful compactification of a symmetric space $G/H$. Take a $\widetilde G$--equivariant principal $R$--bundle $E$ on $M$, where $R$ is a complex reductive algebraic group and $\widetilde G$ is the universal cover of $G$. If the action of the isotropy group $\widetilde H$ on the fiber of $E$ at the identity coset is irreducible, then we prove that $E$ is polystable with respect to any polarization on $M$. Further, for wonderful compactification of the quotient of $\text{PSL}(n,{\mathbb C})$, $n\,\neq\, 4$ (respectively, $\text{PSL}(2n,{\mathbb C})$, $n \geq 2$) by the normalizer of the projective orthogonal group (respectively, the projective symplectic group), we prove that the tangent bundle is stable with respect to any polarization on the wonderful compactification.