Perverse sheaves and $t$-structures on the thin and thick affine flag varieties

Abstract

We study the categories ${\mathrm{Perv}}_{\mathrm{thin}}$, ${\mathrm{Perv}}_{\mathrm{thick}}$ of Iwahori-equivariant perverse sheaves on the thin and thick affine flag varieties associated to a split reductive group $G$. An earlier work of the first author describes ${\mathrm{Perv}}_{\mathrm{thin}}$ in terms of bimodules over the so-called noncommutative Springer resolution. We partly extend this result to ${\mathrm{Perv}}_{\mathrm{thick}}$, providing a similar description for its anti-spherical quotient. The long intertwining functor realizes ${\mathrm{Perv}}_{\mathrm{thick}}$ as the Ringel dual of ${\mathrm{Perv}}_{\mathrm{thin}}$; we point out that it shares some exactness properties with the similar functor acting on perverse sheaves on the finite-dimensional flag variety. We use this result to resolve a conjecture of Arkhipov and the first author, proving that the image in the Iwahori–Whittaker category of any convolution-exact perverse sheaf on the affine flag variety is tilting.