Drinfeld Functor and Finite-Dimensional Representations of Yangian

We extend the results of Drinfeld on Drinfeld functor to the case l>n. We present the character of finite-dimensional representations of the Yangian Y(sl_n) in terms of the Kazhdan-Lusztig polynomials as a consequence.Comment: Latex2e, 17pages, corrected typo

Rationality of admissible affine vertex algebras in the category O

We study the vertex algebras associated with modular invariant representations of affine Kac-Moody algebras at fractional levels, whose simple highest weight modules are classified by Joseph's characteristic varieties. We show that an irreducible highest weight representation of a non-twisted affine Kac-Moody algebra at an admissible level k is a module over the associated simple affine vertex algebra if and only if it is an admissible representation whose integral root system is isomorphic to that of the vertex algebra itself. This in particular proves the conjecture of Adamovic and Milas on the rationality of admissible affine vertex algebras in the category O.Comment: Improved exposition, to appear in Duke Math.

W-algebras at the critical level

Let g be a complex simple Lie algebra, f a nilpotent element of g. We show that (1) the center of the W-algebra $W^{cri}(g,f)$ associated with (g,f) at the critical level coincides with the Feigin-Frenkel center of the affine Lie algebra associated with g, (2) the centerless quotient $W_{\chi}(g,f)$ of $W^{cri}(g,f)$ corresponding to an oper $\chi$ on the disc is simple, (3) the simple quotient $W_{\chi}(g,f)$ is a quantization of the jet scheme of the intersection of the Slodowy slice at f with the nilpotent cone of g

Two-sided BGG resolutions of admissible representations

We prove the conjecture of Frenkel, Kac and Wakimoto on the existence of two-sided BGG resolutions of G-integrable admissible representations of affine Kac-Moody algebras at fractional levels. As an application we establish the semi-infintie analogue of the generalized Borel-Weil theorem for mimimal parabolic subalgebras which enables an inductive study of admissible representations.Comment: revised, to appear in Representation Theor