On the variety of a highest weight module

Goldie rank in the enveloping algebra of a semisimple Lie algebra, III

— Let U (9) be the enveloping algebra of a complex semisimple Lie algebra 9. Many questions concerning U (9) (for example the ordering [14] in Prim U (9)) can be answered in terms of the multiplicities in the composition series of the Verma modules. The main result of this paper shows that a conjecture of Jantzen ([9], 5.18) concerning a certain filtration of the Verma modules implies precisely the formula for these multiplicities which was recently conjectured by Kazhdan and Lusztig ([10], Conj. 1.5). Its proof requires a study of the appropriate generalizations of the so-called Bernstein-Gelfand-Gelfand (9 category which arise when the base field is replaced by a commutative ring (denned over the integers). Several results on the corresponding extension groups are also obtained.

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Towards the Kazhdan-Lusztig conjecture

This is a write-up of the lectures given by the author during the Master Class "Categorification" at Arhus University, Denmark in October 2010.

Lectures on algebraic categorification

Some problems on infinite dimensional lie algebras and their representations

Let 9 be a complex semisimple Lie algebra. In ([8], Prop. 12) Duflo gave a remarkable sum formula interrelating induced ideals. The main result of this paper provides a natural generalization of this formula and more precisely gives a resolution for certain primitive quotients of the enveloping algebra U (g). The proof has three distinct steps. One, the extension of the Bernstein-Gelfand-Gelfand (in short, B.G.G.) resolution of a simple finite dimensional U(g) module to certain simple highest weight modules. Two, the description of the so-called t-finite part of the space of homomorphisms of any one Verma module to any other. Three, the proof of exactness of a certain functor. The last can be viewed as a non-trivial generalization of the fact that a Verma module with dominant highest weight is projective in the so-called C category. A by-product gives some results on a problem of Kostant relating U (g) to the t-finite part of the space of endomorphisms of a simple highest weight module.

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O. Gabber

Papers

On the Bernstein-Gelfand-Gelfand resolution and the Duflo sum formula