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

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.