On the characteristic map of finite unitary groups

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 one to analyze the character theory of the general linear group via symmetric group combinatorics. Using works 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 from this we deduce various character theoretic results. Applications include calculating certain sums of character degrees, and giving 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.