TheN-Eigenvalue Problem and Two Applications

Consider the following two questions. (1) When does a compact Lie group G ⊂ U(n) have an element g ∈ G possessing exactly two eigenvalues? (2) When does a compact Lie group G ⊂ U(n) have a cocharacter U(1) → G such that the compositionU(1) → U(n) is a representation ofU(1)with exactly two weights? A solution to the second problem gives a family of solutions to the first, by choosing g to be almost any element of the image of U(1). The converse is not true. For one thing, any noncentral element of order 2 in G has exactly two eigenvalues. To eliminate these essentially trivial solutions,we can insist that the ratio between the two eigenvalues is not −1. There remain interesting cases of finite groups G satisfying the first (but obviously not the second) condition, especially when the ratio of eigenvalues is a third or fourth root of unity (see [3, 18, 35] for classification results). On the other hand,when G is infinite modulo center, the solutions of the two problems are essentially the same, though the historical reasons for considering them were quite different. The first problem was recently solved in the infinite-mod-center case by Freedman, Larsen, and Wang [13] with an eye toward understanding representations of Hecke algebras. The second problem was solved by Serre [27] nearly thirty years ago in order to classify representations arising from Hodge-Tate modules of weight 1.