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) over tilde -equivariant principal R-bundle Eon M, where R is a complex reductive algebraic group and a is the universal cover of (G) over tilde. If the action of the isotropy group (H) over tilde 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 not equal 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. (C) 2014 Elsevier Inc. All rights reserved.