On Springer correspondence for wreath products

We first show that the wreath product ₘ d 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 ₘ d, via the Bialynicki-Birula decomposition. Furthermore, we realize the group algebra Qₘ d as the top Borel-Moore homology of a Steinberg variety. Such a geometric realization leads to a Springer correspondence for the irreducible representations over Cₘ d, which can be regarded as a counterpart of the Clifford theory for wreath products.