DYNKIN DIAGRAMS AND ENVELOPING-ALGEBRAS OF SEMISIMPLE LIE-ALGEBRAS

Let g be a semi-simple complex Lie algebra, U = U(g) its enveloping al gebra, and A a minimal primitive factor of U, with central character c hi. Under the assumption that chi is regular and integral, we prove th at the Dynkin diagram of g is a Morita invariant of A. Further, a slig ht refinement implies that the flag variety of g is determined, within all generalized flag varieties, by its ring of differential operators . Then we derive the following consequences. First, if U(g) congruent to U(g'), for some Lie algebra g', then g' congruent to g. Second, any automorphism of U acts on the centre, and on some dense open subset o f the primitive spectrum, as a diagram automorphism. We conjecture tha t this result holds true on the whole primitive spectrum, and give K-t heoretic versions of the result and the conjecture. We also improve a key result of [1]. Finally, when chi is only assumed to be regular, we prove, using a result of Soergel, that the Weyl group of g is a Morit a invariant of A. (C) Elsevier, Paris.