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
.