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Register a new userThe symmetric invariants of centralizers and Slodowy grading II
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Let $mathfrak{g}$ be a finite-dimensional simple Lie algebra of rank $ell$ over an algebraically closed field $Bbbk$ of characteristic zero, and let $(e,h,f)$ be an $mathfrak{sl}_2$-triple of g. Denote by $mathfrak{g}^{e}$ the centralizer of $e$ in $mathfrak{g}$ and by ${rm S}(mathfrak{g}^{e})^{mathfrak{g}^{e}}$ the algebra of symmetric invariants of $mathfrak{g}^{e}$. We say that $e$ is good if the nullvariety of some $ell$ homogenous elements of ${rm S}(mathfrak{g}^{e})^{mathfrak{g}^{e}}$ in $(mathfrak{g}^{e})^{*}$ has codimension $ell$. If $e$ is good then ${rm S}(mathfrak{g}^{e})^{mathfrak{g}^{e}}$ is a polynomial algebra. In this paper, we prove that the converse of the main result of arXiv:1309.6993 is true. Namely, we prove that $e$ is good if and only if for some homogenous generating sequence $q_1,ldots,q_ell$, the initial homogenous components of their restrictions to $e+mathfrak{g}^{f}$ are algebraically independent over $Bbbk$.
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Let g be a finite-dimensional simple Lie algebra of rank r over an algebraically closed field of characteristic zero, and let e be a nilpotent element of g. Denote by g^e the centralizer of e in g and by S(g^e)^{g^e} the algebra of symmetric invariants of g^e. We say that e is good if the nullvariety of some r homogeneous elements of S(g^e)^{g^e} in the dual of g^{e} has codimension r. If e is good then S(g^e)^{g^e} is polynomial. The main result of this paper stipulates that if for some homogeneous generators of S(g^e)^{g^e}, the initial homogeneous component of their restrictions to e+g^f are algebraically independent, with (e,h,f) an sl2-triple of g, then e is good. As applications, we obtain new examples of nilpotent elements that verify the above polynomiality condition, in in simple Lie algebras of both classical and exceptional types. We also give a counter-example in type D_7.
Every conic symplectic singularity admits a universal Poisson deformation and a universal filtered quantization, thanks to the work of Losev and Namikawa. We begin this paper by showing that every such variety admits a universal equivariant Poisson deformation and universal equivariant quantization with respect to any group acting on it by $mathbb{C}^times$-equivariant Poisson automorphisms. We go on to study these definitions in the context of nilpotent Slodowy slices. First we give a complete description of the cases in which the finite $W$-algebra is the universal filtered quantization of the slice, building on the work of Lehn--Namikawa--Sorger. This leads to a near-complete classification of the filtered quantizations of nilpotent Slodowy slices. The subregular slices in non-simply-laced Lie algebras are especially interesting: with some minor restrictions on Dynkin type we prove that the finite $W$-algebra is the universal equivariant quantization with respect to the Dynkin automorphisms coming from the unfolding of the Dynkin diagram. This can be seen as a non-commutative analogue of Slodowys theorem. Finally we apply this result to give a presentation of the subregular finite $W$-algebra in type B as a quotient of a shifted Yangian.
same as arXiv:0904.1778
For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.
Suppose V is a finite dimensional, complex vector space, A is a finite set of codimension one subspaces of V, and G is a finite subgroup of the general linear group GL(V) that permutes the hyperplanes in A. In this paper we study invariants and semi-invariants in the graded QG-module H^*(M(A)), where M(A) denotes the complement in V of the hyperplanes in A and H^* denotes rational singular cohomology, in the case when G is generated by reflections in V and A is the set of reflecting hyperplanes determined by G, or a closely related arrangement. Our main result is the construction of an explicit, natural (from the point of view of Coxeter groups) basis of the space of invariants, H^*(M(A))^G. In addition to leading to proof of the description of the space of invariants conjectured by Felder and Veselov for Coxeter groups that does not rely on computer calculations, this construction provides an extension of the description of the space of invariants proposed by Felder and Veselov to arbitrary finite unitary reflection groups.
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