Lagrangian Construction of the (gl N , Gl M )-duality

We give a geometric realization of the symmetric algebra of the tensor space C ⊗ C m together with the action of the dual pair (gln, glm) in terms of lagrangian cycles in the cotangent bundles of certain varieties. We establish geometrically the equivalence between the (gln, glm)-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. 0 Introduction The left action of gln (resp. glm) on the first (resp. second) factor of the tensor product C ⊗ C m induces natural left actions on the d-th symmetric tensor space S(C ⊗ C ). These two actions clearly commute with each other and they form a dual pair in the sense of Howe (cf. [H]). Following Howe, we have the isotypic decomposition S(C ⊗ C ) = ⊕ λ∈P min(n,m) Vλ ⊗ Vλ (1) where P k is the set of partitions of d into at most k parts and Vλ (resp. Vλ) is the irreducible module of gln (resp. glm) with highest weight λ, and min(n,m) denotes the minimum of n and m. On the other hand, we consider the set F of n-step flags F = (0 = F0 ⊂ . . . ⊂ Fn = C ) of the vector space C of complex dimension d, with the induced action of the general linear group GLd. In such a setup, Beilinson, Lusztig and MacPherson [BLM] constructed the quantum group for gln. Inspired by their construction, Ginzburg [G] obtained a micro-local version of it for the enveloping algebra U(gln) in terms of lagrangian cycles in the cotangent bundles of F × F . This is an analog of the Springer theory for Weyl groups (cf. e.g. [CG, Hu]). Note that statements are made in [G, CG] in terms of sln although the construction indeed yields gln. For our purpose, it is important to stick to gln. Let N be the nilpotent cone in the general linear Lie algebra gld and let Nn be the subset of n-step nilpotents in N . Let M be the set M := {(x, F) ∈ Nn × F | x(Fi) ⊂ Fi−1, i = 1, . . . , n}. 1