Quantization of Slodowy slices (Q2778022)

Let \(\mathfrak g\) be a complex semisimple Lie algebra, \(\mathbb{O}\) a nonzero nilpotent \(\operatorname {Ad}G\)-orbit in \(\mathfrak g\), and \(e\in \mathbb{O}\). There is an \({\mathfrak {sl}}_2\)-triple \((e,f,h)\) associated to \(e\). The Slodowy slice to \(\mathbb{O}\) at \(e\) is defined to be the affine space \(e+\ker\operatorname {ad} f\). It is known that there is an isomorphism \(\Phi: \mathfrak g \rightarrow \mathfrak g^*\) such that \(\langle \Phi(e), f\rangle =1\). Let \(\chi=\Phi(e)\) and \(\mathcal{S}=\Phi(e\ker\) ad \(f)\). The authors show that \(\mathcal{S}\) has a natural Poisson structure. The aim of this paper is to construct a quantization of \(\mathcal{S}\). Fixing an isotropic subspace \(l\) of \(\mathfrak g(-1)=\{x\in {\mathfrak g}\mid [h,x]=-x\}\), they define a special subspace \(\mathcal{H}_l\) of \(\mathbb{O}_l\), the induced \(U{\mathfrak g}\)-module. Then they define the Kazhdan grading on \(\mathbb{C}[\mathcal{S}]\) and the Kazhdan filtration on \(\mathcal{H}_l\). It is shown that gr \(\mathcal{H}_l\) is canonically isomorphic to \(\mathbb{C}[\mathcal{S}]\) as graded Poisson algebras and that \(\mathcal{H}_l\) is independent of the choice of \(l\).