, Volume 56, Issue 3, pp 191-213

Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture

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Abstract

Let g be a complex semisimple Lie algebra andU(g) its enveloping algebra. GivenM a simpleU(g) module, letL(M, M) denote the subspace of ad g finite elements of Hom(M, M). Kostant has asked if the natural homomorphism ofU(g) intoL(M, M) is surjective. Here the question is analysed for simple modules with a highest weight vector. This has a negative answer if g admits roots of different length ([7], 6.5). Here general conditions are obtained under whichU(g)/AnnM andL(M, M) have the same ring of fractions—in particular this is shown to always hold if g has only typeA n factors. Combined with [21], this provides a method for determining the Goldie ranks for the primitive quotients ofU(g). Their precise form is given in typeA n (Cartan notation) for which the generalized Gelfand-Kirillov conjecture for primitive quotients is also established.

This paper was written while the author was a guest of the Institute for Advanced Studies, the Hebrew University of Jerusalem and on leave of absence from the Centre Nationale de la Recherche Scientifique