  In his classic book on symmetric functions, Macdonald describes a remarkable
result by Green relating the character theory of the finite general linear
group to transition matrices between bases of symmetric functions. This
connection allows us to analyze the representation theory of the general linear
group via symmetric group combinatorics. Using the work of Ennola, Kawanaka,
Lusztig and Srinivasan, this paper describes the analogous setting for the
finite unitary group. In particular, we explain the connection between
Deligne-Lusztig theory and Ennola's efforts to generalize Green's work, and
deduce various representation theoretic results from these results.
Applications include finding certain sums of character degrees, and a model of
Deligne-Lusztig type for the finite unitary group, which parallels results of
Klyachko and Inglis and Saxl for the finite general linear group.
