Derived subalgebras of centralisers and finite W-algebras

Abstract

Let g = Lie(G) be the Lie algebra of a simple algebraic group G over an algebraically closed field of characteristic 0. Let e be a nilpotent element of g and let g_e = Lie(G_e) where G_e stands for the stabiliser of e in G. For g classical, we give an explicit combinatorial formula for the codimension of [g_e, g_e] in g_e and use it to determine those e in g for which the largest commutative quotient U(g,e)^{ab} of the finite W-algebra U(g,e) is isomorphic to a polynomial algebra. It turns out that this happens if and only if e lies in a unique sheet of g. The nilpotent elements with this property are called non-singular in the paper. Confirming a recent conjecture of Izosimov we prove that a nilpotent element e in g is non-singular if and only if the maximal dimension of the geometric quotients S/G, where S is a sheet of g containing e, coincides with the codimension of [g_e,g_e] in g_e and describe all non-singular nilpotent elements in terms of partitions. We also show that for any nilpotent element e in a classical Lie algebra g the closed subset of Specm U(g,e)^{ab} consisting of all points fixed by the natural action of the component group of G_e is isomorphic to an affine space. Analogues of these results for exceptional Lie algebras are also obtained and applications to the theory of primitive ideals are given.

arXiv comment66 pages, 7 tables, 2 figures; the present version has been accepted for publication

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• Induced nilpotent orbits of the simple Lie algebras of exceptional type
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Cited by in corpus (5)

• Rigid orbits and sheets in reductive Lie algebras over fields of prime characteristic
• Minimal dimensional representations of reduced enveloping algebras for ${\mathfrak{gl}}_n$
• Atomic Higgsings of 6D SCFTs II: Induced Flows
• Whittaker categories and the minimal nilpotent finite $W$-algebra for ${\mathfrak{sl}}_{n+1}$
• Multiplicity-free primitive ideals associated with rigid nilpotent orbits