Abstract
We study the ring of sections A ( X ) of a complete symmetric variety X , that is of the wonderful completion of G / H where G is an adjoint semisimple group and H is the fixed subgroup for an involutorial automorphism of G . We find generators for Pic( X ), we generalize the PRV conjecture to complete symmetric varieties and construct a standard monomial theory for A ( X ) that is compatible with G orbit closures in X . This gives a degeneration result and the rational singularityness for A ( X ).
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