On the Springer correspondence for wreath products

Abstract

We establish a Bruhat decomposition indexed by the wreath product $\hat{I}{\pounds }_m\wr \hat{I}{\pounds }_d$ between two symmetric groups -- note that $\hat{I}{\pounds }_m\wr \hat{I}{\pounds }_d$ 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 ${\mathbb{C}}^{\ast }$-actions. Next, we construct a Steinberg variety whose top Borel-Moore homology realizes the group algebra $\mathbb{Q}[\hat{I}{\pounds }_m\wr \hat{I}{\pounds }_d]$ as a proper subalgebra. Such a geometric realization leads to a Springer-type correspondence which identifies the irreducible representations of $\hat{I}{\pounds }_m\wr \hat{I}{\pounds }_d$ 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.

26 pages. v4: to appear in JPAA. Expositions improved. v3: Stronger results are proved on the (classification of) simple modules to address and correct a flaw identified in the previous version