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Classification of finite-dimensional irreducible modules over $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}W$-algebras

Part of: Lie algebras and Lie superalgebras Representation theory of rings and algebras

Published online by Cambridge University Press:  07 April 2014

Ivan Losev  and
Victor Ostrik
Ivan Losev
Affiliation:
Department of Mathematics, Northeastern University, Boston, MA 02115, USA email [email protected]
Victor Ostrik
Affiliation:
Department of Mathematics, University of Oregon, Eugene, OR 97403, USA email [email protected]
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Abstract

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Finite $W$-algebras are certain associative algebras arising in Lie theory. Each $W$-algebra is constructed from a pair of a semisimple Lie algebra ${\mathfrak{g}}$ (our base field is algebraically closed and of characteristic 0) and its nilpotent element $e$. In this paper we classify finite-dimensional irreducible modules with integral central character over $W$-algebras. In more detail, in a previous paper the first author proved that the component group $A(e)$ of the centralizer of the nilpotent element under consideration acts on the set of finite-dimensional irreducible modules over the $W$-algebra and the quotient set is naturally identified with the set of primitive ideals in $U({\mathfrak{g}})$ whose associated variety is the closure of the adjoint orbit of $e$. In this paper, for a given primitive ideal with integral central character, we compute the corresponding $A(e)$-orbit. The answer is that the stabilizer of that orbit is basically a subgroup of $A(e)$ introduced by G. Lusztig. In the proof we use a variety of different ingredients: the structure theory of primitive ideals and Harish-Chandra bimodules for semisimple Lie algebras, the representation theory of $W$-algebras, the structure theory of cells and Springer representations, and multi-fusion monoidal categories.

Keywords

W-algebrasemisimple Lie algebranilpotent orbitprimitive idealHarish-Chandra bimodulecells in Weyl groupsSpringer representationLusztig’s subgroupsasymptotic Hecke algebra

MSC classification

Secondary: 16G99: None of the above, but in this section 17B35: Universal enveloping (super)algebras
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 6 , June 2014 , pp. 1024 - 1076
Copyright
© The Author(s) 2014 

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