On Ginzburg's Lagrangian construction of representations of GL(n)

Abstract

In [3] V. Ginzburg observed that one can realize irreducible representations of the group GL(n, ℂ) in the cohomology of certain Springer's fibers for the group GLd (for all d ∈ ℕ). However, Ginzburg's construction of the action of GL(n) on this cohomology was a bit artificial (he defined the action of Chevalley generators of the Lie algebra gln on the corresponding cohomology by certain explicit correspondences, following the work of A. Beilinson, G. Lusztig and R. MacPherson ([1]), who gave a similar construction of the quantum group Uq(gln)). In this note we give a very simple geometric definition of the action of the whole group GL(n, ℂ) on the above cohomology and simplify the results of [3].