W-algebras at the critical level

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 $W_χ(g,f)$ of $W^{cri}(g,f)$ corresponding to an oper $χ$ on the disc is simple, (3) the simple quotient $W_χ(g,f)$ is a quantization of the jet scheme of the intersection of the Slodowy slice at f with the nilpotent cone of g.

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