Iwahori-equivariant convolution conjecture for the affine Grassmannian

Let $G$ be a semisimple group, let $I\subset L^+G$ be an Iwahori subgroup, and let $Gr$ be the affine Grassmannian. Let $P_I(Gr)$ be the category of $I$-equivariant perverse sheaves on $Gr$ with compact support, and let $P(Gr)$ be the category of spherical perverse sheaves. Convolution is denoted by $\ast$. The category $P_I(Gr)$ is non-semisimple and contains $P(Gr)$ as a subcategory. Iwahori-equivariant convolution conjecture. (i) For any $M\in P_I(Gr)$ and $A\in P(Gr)$, $M\ast A\in P_I(Gr)$, yielding a bifunctor $\ast:P_I(Gr)\times P(Gr)\to P_I(Gr)$. (ii) Via the Kazhdan–Lusztig–Kashiwara–Tanisaki equivalence, this bifunctor corresponds to the standard tensor product of representations of the quantum group. (iii) In particular, for fixed $A\in P(Gr)$, the functor $M\mapsto M\ast A$ on the non-semisimple category $P_I(Gr)$ is exact. This conjecture is proposed as a generalization of theorems 1.3.1 and 1.4.1 and is related to the quantum-group interpretation of intersection cohomology.