On equivariant principal bundles over wonderful compactifications

Let G be a simple algebraic group of adjoint type over C, and let M be the wonderful compactification of a symmetric space G/H. Take a G˜-equivariant principal R-bundle E on M, where R is a complex reductive algebraic group and G˜ is the universal cover of G. If the action of the isotropy group 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 PSL(n,C), n≠4 (respectively, PSL(2n,C), n>1) 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.