On Springer correspondence for wreath products

We first show that the wreath product Σₘ Σd between two symmetric groups appears as the generalized Weyl group of an Iwahori's generalized Tits system. We then introduce a certain subvariety of the flag variety of type A, and then give a geometric proof of its Bruhat decomposition indexed by Σₘ Σd, via the Bialynicki-Birula decomposition. Furthermore, we realize the group algebra Q[Σₘ Σd] as the top Borel-Moore homology of a Steinberg variety. Such a geometric realization leads to a Springer correspondence for the irreducible representations over C[Σₘ Σd], which can be regarded as a counterpart of the Clifford theory for wreath products.