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Register a new userCenter at the critical level for centralizers in type $A$
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A. I. Molev
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We consider the affine vertex algebra at the critical level associated with the centralizer of a nilpotent element in the Lie algebra $mathfrak{gl}_N$. Due to a recent result of Arakawa and Premet, the center of this vertex algebra is an algebra of polynomials. We construct a family of free generators of the center in an explicit form. As a corollary, we obtain generators of the corresponding quantum shift of argument subalgebras and recover free generators of the center of the universal enveloping algebra of the centralizer produced earlier by Brown and Brundan.
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We introduce a new family of affine $W$-algebras associated with the centralizers of arbitrary nilpotent elements in $mathfrak{gl}_N$. We define them by using a version of the BRST complex of the quantum Drinfeld--Sokolov reduction. A family of free generators of the new algebras is produced in an explicit form. We also give an analogue of the Fateev--Lukyanov realization for these algebras by applying a Miura-type map.
In arXiv:1001.2562 a certain non-commutative algebra $A$ was defined starting from a semi-simple algebraic group, so that the derived category of $A$-modules is equivalent to the derived category of coherent sheaves on the Springer (or Grothendieck-Springer) resolution. Let $hat{g}$ be the affine Lie algebra corresponding to the Langlands dual Lie algebra. Using results of Frenkel and Gaitsgory arXiv:0712.0788 we show that the category of $hat{g}$ modules at the critical level which are Iwahori integrable and have a fixed central character, is equivalent to the category of modules over a quotient of $A$ by a central character. This implies that numerics of Iwahori integrable modules at the critical level is governed by the canonical basis in the $K$-group of a Springer fiber, which was conjecturally described by Lusztig and constructed in arXiv:1001.2562.
We introduce a new family of Poisson vertex algebras $mathcal{W}(mathfrak{a})$ analogous to the classical $mathcal{W}$-algebras. The algebra $mathcal{W}(mathfrak{a})$ is associated with the centralizer $mathfrak{a}$ of an arbitrary nilpotent element in $mathfrak{gl}_N$. We show that $mathcal{W}(mathfrak{a})$ is an algebra of polynomials in infinitely many variables and produce its free generators in an explicit form. This implies that $mathcal{W}(mathfrak{a})$ is isomorphic to the center at the critical level of the affine vertex algebra associated with $mathfrak{a}$.
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.
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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