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 double-struck 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(double-struck 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.