W-algebras at the critical level

Abstract

Let g be a complex simple Lie algebra, f a nilpotent element of g. We show that (1) the center of the W-algebra $W^{cri}(g,f)$ associated with (g,f) at the critical level coincides with the Feigin-Frenkel center of the affine Lie algebra associated with g, (2) the centerless quotient $W_{\chi }(g,f)$ of $W^{cri}(g,f)$ corresponding to an oper $\chi$ on the disc is simple, (3) the simple quotient $W_{\chi }(g,f)$ is a quantization of the jet scheme of the intersection of the Slodowy slice at f with the nilpotent cone of g.

References in corpus (7)

• Vertex Algebras and Algebraic Curves
• Quantum Reduction for Affine Superalgebras
• Jet Schemes of Locally Complete Intersection Canonical Singularities
• Quantization of Slodowy slices
• Representation Theory of W-Algebras
• Harish-Chandra bimodules for quantized Slodowy slices
• On the restricted Verma modules at the critical level

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• $\mathcal{G}$-systems
• Newton--Okounkov bodies and minimal models for cluster varieties