Pointwise purity, derived Satake, and Symplectic duality

It has been known for a long time that Ext's between IC-sheaves may often be expressed in terms of Hom's between cohomology groups. We prove a more general result under weaker assumptions. The result is used to describe the action of the derived Satake equivalence on !-pure objects and show that the equivalence enjoys a new kind of functoriality with respect to morphisms of reductive groups. We find and prove normality of the symplectic dual X^! for many smooth affine Hamiltonian G-varieties X, including X=T^*(G/H) for all connected reductive subgroups H of G. We also describe the symplectic duals M^! in the case of Coulomb branches and prove that M^! has symplectic singularities.