Lagrangian construction of the $(gl_n, gl_m)$-duality

We give a geometric realization of the symmetric algebra of the tensor space $C^n \bigotimes C^m$ 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 $A$.