On special pieces, the Springer correspondence, and unipotent characters

Let $G$ be a connected reductive algebraic group over the algebraic closure of a finite field ${\Bbb F}_q$ of good characteristic. In this paper, we demonstrate a remarkable compatibility between the Springer correspondence for $G$ and the parametrization of unipotent characters of $G({\Bbb F}_q)$. In particular, we show that in a suitable sense, ``large'' portions of these two assignments in fact coincide. This extends earlier work of Lusztig on Springer representations within special pieces of the unipotent variety.