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

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

1999-07-23

arxiv.org/abs/math/9907154v2

Commun. Contemp. Math. 3 (2001) 201--214

Mathematics

Representation Theory

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

Scientific paper

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 $A$.

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