  The Glauberman correspondence is a fundamental bijection in the character
theory of finite groups. In 1994, Hartley and Turull established a
degree-divisibility property for characters related by that correspondence,
subject to a congruence condition which should hold for the Green functions of
finite groups of Lie type, as defined by Deligne and Lusztig. Here, we present
a general argument for completing the proof of that congruence condition.
Consequently, the degree-divisibility property holds in complete generality.
