On the Springer correspondence for wreath products

Abstract

We establish a Bruhat decomposition indexed by the wreath product $Σ_m\wr Σ_d$ between two symmetric groups -- note that $Σ_m\wr Σ_d$ is not a Coxeter group in general. We show that such a decomposition affords a geometric variant in terms of the Bialynicki-Birula decomposition for varieties with $\mathbb{C}^*$-actions. Next, we construct a Steinberg variety whose top Borel-Moore homology realizes the group algebra $\mathbb{Q}[Σ_m\wr Σ_d]$ as a proper subalgebra. Such a geometric realization leads to a Springer-type correspondence which identifies the irreducible representations of $Σ_m\wr Σ_d$ with isotypic components of certain unconventional Springer fibers using type A geometry. In other words, we obtain a geometric counterpart of the (algebraic) Clifford theory, for the first time. Consequently, we obtain a new Springer correspondence of Weyl groups of type B/C/D using essentially type A geometry.