We study the categories $\mathrm{ Perv}_{ \mathrm{ thin}}$ and $\mathrm{ Perv}_{ \mathrm{ thick}}$ 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 non-commutative 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.
Categories: math.RT math.AG