Remarks on the wonderful compactification of semisimple algebraic groups

We prove that if G is a semisimple algebraic group of adjoint type over the field of complex numbers, H is the subgroup of all fixed points of an involution sigma of G that is induced by an involution <(sigma)over cap> of the simply connected covering (G) over cap of G, then the wonderful compactification G/H of the homogeneous space G/H is isomorphic to the G.I.T quotient (G) over bar(ss)(L)//H of the wonderful compactification (G) over bar of G for a suitable choice of a line bundle L on (G) over bar. We also prove a functorial property of the wonderful compactifications of semisimple algebraic groups of adjoint type.