Character sheaves and generalized Springer correspondence

Let G be a connected reductive algebraic group over an algebraic closure of a finite field of characteristic p. Under the assumption that p is good for G, we prove that for each character sheaf A on G which has non-zero restriction to the unipotent variety of G, there exists a unipotent class C-A canonically attached to A, such that A has non-zero restriction on C-A, and any unipotent class C in G on which A has non-zero restriction has dimension strictly smaller than that of C-A.