Commutative quotients of finite \(W\)-algebras (Q986078)

This paper is a continuation of the author's study of finite \(W\)-algebras, started in [\textit{A. Premet}, Mosc. Math. J. 7, No. 4, 743--762 (2007; Zbl 1139.17005)]. Let \(G\) be a simple, simply connected algebraic group over \(\mathbb{C}\), and \(\mathfrak{g}\) be its Lie algebra. Let \((e,f,h)\) be an \(\mathfrak{sl}_2\)-triple in \(\mathfrak{g}\), and \(U(\mathfrak{g},e)\) be the corresponding finite \(W\)-algebra. Three main topics arinvestigated. First, the author studies a relationship between representations of \(U(\mathfrak{g},e)\) and modular non-restricted representations of \(\mathfrak{g}_{k}\), where \(k\) is a field of characteristic \(p > 0\). The main result in this direction is as follows. Let \(\chi\) be the nilpotent character corresponding to \(e\), and \(\mathcal{O}_{\chi}\) be the coadjoint orbit through \(\chi\). If \(U(\mathfrak{g},e)\) affords a one-dimensional representation, then the reduced enveloping algebra \(U_{\chi}(\mathfrak{g}_k)\) has a simple representation of dimension \(p^{\frac{1}{2}\dim\, \mathcal{O}_{\chi}}\). As a corollary, a solution to the problem of small representations is presented for simple Lie algebras of types \(B-D\) in characteristic \(p \gg 0\). Second, sheets and commutative quotients of \(U(\mathfrak{g},e)\) are described. Let \(\mathcal{O}\) be the adjoint orbit through \(e\). Among other striking results, the authors computes the Krull dimension of \(U(\mathfrak{g},e)^{\mathrm{ab}}\), the largest commutative quotient of \(U(\mathfrak{g},e)\), in the case when \(\mathcal{O}\) is not rigid. When \(\mathcal{O}\) is rigid, the maximal spectrum of \(U(\mathfrak{g},e)^{\mathrm{ab}}\) is shown to be a finite set (possibly empty). As an application, an explicit description of \(U(\mathfrak{g},e)^{\mathrm{ab}}\) is given for \(\mathfrak{g} = \mathfrak{gl}_n\) or \(\mathfrak{sl}_n\). Third, the author provides a generalised Gelfand-Graev model for primitive ideals in \(U(\mathfrak{g},e)\). A similar construction was given in [\textit{A. Premet} (loc. cit.)] under the technical assumption that the ideal is rational. The result was proved in full generality with different methods in [\textit{I. Losev}, J. Am. Math. Soc. 23, No. 1, 35--59 (2010; Zbl 1246.17015)] and [\textit{V. Ginzburg}, Represent. Theory 13, 236--271 (2009; Zbl 1250.17007)]. As a consequence of the result, the author shows that for any complex semisimple Lie algebra \(\mathfrak{g}\), the primitive spectrum of \(U(\mathfrak{g})\) is a countable disjoint union of quasi-affine algebraic varieties. This is a solution to a problem of Borho and Dixmier, see [\textit{J. Dixmier}, Algèbres enveloppantes. Paris etc.: Gauthier-Villars (1974; Zbl 0308.17007)].