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

We study the categories Pervₜₕᵢₙ , Pervthick 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 Pervₜₕᵢₙ in terms of bimodules over the so-called noncommutative Springer resolution. We partly extend this result to Pervthick , providing a similar description for its anti-spherical quotient. The long intertwining functor realizes Pervthick as the Ringel dual of Pervₜₕᵢₙ ; 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.