The torus equivariant cohomology rings of Springer varieties

The Springer variety of type A   associated to a nilpotent operator on CnCn in Jordan canonical form admits a natural action of the ℓ  -dimensional torus TℓTℓ where ℓ   is the number of the Jordan blocks. We give a presentation of the TℓTℓ-equivariant cohomology ring of the Springer variety through an explicit construction of an action of the n  -th symmetric group on the TℓTℓ-equivariant cohomology. The TℓTℓ-equivariant analogue of so-called Tanisaki's ideal will appear in the presentation.