Higher real K-theories and topological automorphic forms

Given a maximal finite subgroup G of the nth Morava stabilizer group at a prime p, we address the question: is the associated higher real K-theory EOn a summand of the K(n)-localization of a TAF-spectrum associated to a unitary similitude group of type U(1, n − 1)? We answer this question in the affirmative for p ∈ {2, 3, 5, 7} and n = (p − 1)pr−1 for a maximal finite subgroup containing an element of order p. We answer the question in the negative for all other odd primary cases. In all odd primary cases, we give an explicit presentation of a global division algebra with involution in which the group G embeds unitarily.