On the derived Lusztig correspondence

Abstract

Abstract Let G be a connected reductive group, T a maximal torus of G, N the normalizer of T and $W=N/T$ the Weyl group of G. Let ${\mathfrak {g}}$ and ${\mathfrak {t}}$ be the Lie algebras of G and T. The affine variety $\mathfrak {car}={\mathfrak {t}}/\!/W$ of semisimple G-orbits of ${\mathfrak {g}}$ has a natural stratification $$ \begin{align*} \mathfrak{car}=\coprod_{\lambda}\mathfrak{car}_{\lambda} \end{align*} $$ indexed by the set of G-conjugacy classes of Levi subgroups: the open stratum is the set of regular semisimple orbits and the closed stratum is the set of central orbits. In [17], Rider considered the triangulated subcategory $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])^{\mathrm {Spr}}$ of $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\mathrm {nil}}/G])$ generated by the direct summand of the Springer sheaf. She proved that it is equivalent to the derived category of finitely generated dg modules over the smash product algebra ${\overline {\mathbb {Q}}_{\ell }}[W]\# H^{\bullet }_G(G/B)$ where $H^{\bullet }_G(G/B)$ is the G-equivariant cohomology of the flag variety. Notice that the later derived category is $D_{\mathrm {c}}^{\mathrm {b}}(\mathrm {B}(N))$ where $\mathrm {B}(N)=[\mathrm {Spec}(k)/N]$ is the classifying stack of N-torsors. The aim of this paper is to understand geometrically and generalise Rider’s equivalence of categories: For each $\lambda $ we construct a cohomological correspondence inducing an equivalence of categories between $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {t}}_{\lambda }/N])$ and $D_{\mathrm {c}}^{\mathrm {b}}([{\mathfrak {g}}_{\lambda }/G])^{\mathrm {Spr}}$ .