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Finite-dimensional representations of \(W\)-algebras. (English) Zbl 1235.17007

Let \(\mathfrak{g}\) be a finite dimensional Lie algebra over an algebraically closed field of characteristic zero and \(e\in\mathfrak{g}\) be a nilpotent element. Let \(G\) be a simply connected algebraic group with Lie algebra \(\mathfrak{g}\). Let \(\mathbb{O}\) be the adjoint orbit of \(e\). Associated to the pair \((\mathfrak{g},e)\) these is a certain associative algebra \(\mathcal{W}\), called a \(W\)-algebra of finite type. Choose an \(\mathfrak{sl}_2\)-triple \((e,f,h)\) and set \(\mathcal{Q}:=Z_G(e,f,h)\). Let \(\mathcal{Q}^{\circ}\) be the unit component of \(\mathcal{Q}\) and set \(C(e):=\mathcal{Q}/\mathcal{Q}^{\circ}\). In a previous work the author constructed two maps \(I\mapsto I^+\) and \(J\mapsto J_+\) between the sets of two-sided ideals in \(\mathcal{W}\) and \(U(\mathfrak{g})\) with nice properties. The main result of the present paper is the following
Theorem. An ideal \(I\) in \(\mathcal{W}\) of finite codimension equals \(J_+\) for some ideal \(J\) in \(U(\mathfrak{g})\) such that \(V(U(\mathfrak{g})/J)=\overline{\mathbb{O}}\) if and only if \(I\) is \(C(e)\)-invariant. If this is the case, then \(I=(I^+)_+\).
The above theorem implies an almost complete classification of irreducible finite dimensional representations of \(\mathcal{W}\). This classification is complete in some cases, for example, in the case \(\mathfrak{g}=\mathfrak{sl}_n\) (in which it is obtained by Brundan and Kleshchev by completely different methods).
A key ingredient in the proof is a relationship between Harish-Chandra bimodules for \(\mathfrak{g}\) and bimodules over \(\mathcal{W}\).

MSC:

17B35 Universal enveloping (super)algebras
16G30 Representations of orders, lattices, algebras over commutative rings
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