On equivariant principal bundles over wonderful compactifications

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 $\tilde{G}$--equivariant principal $R$--bundle $E$ on $M$, where $R$ is a complex reductive algebraic group and $\tilde{G}$ is the universal cover of $G$. If the action of the isotropy group $\tilde{H}$ 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 $\text{PSL}(n,\mathbb{C})$, $n\ne 4$ (respectively, $\text{PSL}(2n,\mathbb{C})$, $n\ge 2$) 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.