We prove that in characteristic zero the multiplication of sections of line bundles generated by global sections on a complete symmetric variety is a surjective map. As a consequence, the cone defined by a complete linear system over or over a closed -stable subvariety of is normal. This gives an affirmative answer to a question raised by Faltings in . A crucial point of the proof is a combinatorial property of root systems.
"Projective normality of complete symmetric varieties." Duke Math. J. 122 (1) 93 - 123, 15 March 2004. https://doi.org/10.1215/S0012-7094-04-12213-4