The talk is based on work in progress joint with Tina Kanstrup.We recall two representation theoretic realizations of the algebra of symmetric functions: one is via the sum of representation rings of all symmetric groups and the other via the limit of the Grothendieck groups for the categories of polynomial representations of GL(n). The isomorphism of those is explicit and is given by Schur-Weyl correspondence.
Next we recall the Springer correspondence relating representations of a symmetric group S(n) and perverse sheaves on the nilpotent cone. To upgrade the correspondence to an equivalence of derived categories, we recall briefly recent results due to Laura Rider.
Finally we provide a geometric language for representations of GL(n) via the positive part of the affine Grassmannial and upgrade the Schur-Weyl correspondence to a functor that is conjecturally an equivalence of derived categories.
We conclude the talk with an open question of Yan Soybelman that was a motivation for the whole topic.