Lagrangian construction of the $(g{l}_n,g{l}_m)$-duality

Abstract

We give a geometric realization of the symmetric algebra of the tensor space $C^n⨂C^m$ together with the action of the dual pair $(g{l}_n,g{l}_m)$ in terms of lagrangian cycles in the cotangent bundles of certain varieties. We establish geometrically the equivalence between the $(g{l}_n,g{l}_m)$ 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 $A$.

14 pages,latex, minor changes, to appear in Communications in Contemporary Mathematics