  In this paper, we present a conjecture on the degree of unipotent characters
in the cohomology of particular Deligne-Lusztig varieties for groups of Lie
type, and derive consequences of it. These degrees are a necessary piece of
data in the geometric version of Brou\'e's abelian defect group conjecture, and
can be used to verify this geometric conjecture in new cases. The geometric
version of Brou\'e's conjecture should produce a more combinatorially defined
derived equivalence, called a perverse equivalence. We prove that our
conjectural degree is an integer (which is not obvious) and has the correct
parity for a perfect isometry, and verify that it induces a perverse
equivalence for all unipotent blocks of groups of Lie type with cyclic defect
groups, whenever the shape of the Brauer tree is known (i.e., not E7 and E8).
It has also been used to find perverse equivalences for some non-cyclic cases.
This paper is a contribution to the conjectural description of the exact form
of a derived equivalence proving Brou\'e's conjecture for groups of Lie type.
