1-dimensional representations and parabolic induction for W-algebras

A W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. This paper concentrates on the study of 1-dimensional representations of these algebras. Under some conditions on a nilpotent element (satisfied by all rigid elements) we obtain a criterium for a finite dimensional module to have dimension 1. It is stated in terms of the Brundan-Goodwin-Kleshchev highest weight theory. This criterium allows to compute highest weights for certain completely prime primitive ideals in universal enveloping algebras. We make an explicit computation in a special case in type $E_8$. Our second principal result is a version of a parabolic induction for W-algebras. In this case, the parabolic induction is an exact functor between the categories of finite dimensional modules for two different W-algebras. The most important feature of the functor is that it preserves dimensions. In particular, it preserves one-dimensional representations. A closely related result was obtained previously by Premet. We also establish some other properties of the parabolic induction functor.Comment: 31 pages, v2 36 pages, 4 new subsections added, v3 38 pages few gaps fixed, v4 minor changes, v5 references adde

Finite dimensional representations of W-algebras

W-algebras of finite type are certain finitely generated associative algebras closely related to the universal enveloping algebras of semisimple Lie algebras. In this paper we prove a conjecture of Premet that gives an almost complete classification of finite dimensional irreducible modules for W-algebras. Also we study a relation between Harish-Chandra bimodules and bimodules over $W$-algebras.Comment: 19 pages, v2, 21 pages, moderate changes, some mistakes fixed, Corollary 1.3.3 is added, v3 24 pages three new subsections and several remarks added v4 proof of Lemma 2.4.1 expanded, Remark 2.3.2 added, v5 29 pages major changes, v6 more changes, v7 fina

Finite W-algebras

A finite W-algebra is an associative algebra constructed from a semisimple Lie algebra and its nilpotent element. In this survey we review recent developments in the representation theory of W-algebras. We emphasize various interactions between W-algebras and universal enveloping algebras.Comment: The text of a sectional talk for ICM 2010. 21 page