ABSTRACT
Let us return now to the case of a generic untwisted loop algebra C§, built over the simple Lie algebra C§0 (complex). Let C§"' be a real form corresponding to a Hermitian symmetric space,and let w be the conjugation of C§0 with respect to C§"" Consider a Cartan decomposition of C§"': C§"' = tw EB l=lw· Let fJw be a maximal abelian subalgebra oft"'. Then the complexified (f)"')c of fJw is a Cartan subalgebra of C§0. By general results on Cartan decompositions (see [26] for instance), t"' is reductive in C§"'; it has a one-dimensional center <f6"', and one has t"' = t;, EB <f6 00 , t~ = [±00 , iJ.
