  We first show that the wreath product $\Sigma_m\wr \Sigma_d$ 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 $\Sigma_m\wr \Sigma_d$, via the Bialynicki-Birula decomposition.
Furthermore, we realize the group algebra $\mathbb{Q}[\Sigma_m\wr \Sigma_d]$ as
the top Borel-Moore homology of a Steinberg variety. Such a geometric
realization leads to a Springer correspondence for the irreducible
representations over $\mathbb{C}[\Sigma_m\wr \Sigma_d]$, which can be regarded
as a counterpart of the Clifford theory for wreath products. Consequently, we
have obtained a new Springer correspondence of type B/C/D using essentially
type A geometry.
