Lagrangian construction of the (gln,glm)-duality

We give a geometric realization of the symmetric algebra of the tenser space C-n circle times C-m together with the action of the dual pair (gl(n), gl(m)) in terms of lagrangian cycles in the cotangent bundles of certain varieties. We establish geometrically the equivalence between the (gl(n), gl(m))-duality and Schur duality. We establish the connection between Springer's construction of (representations of) Weyl groups and Ginzburg's construction of (representations of) Lie algebras of type A.