  In the 1990's, Bertram, Feinberg and Mukai examined Brill-Noether loci for
vector bundles of rank 2 with fixed canonical determinant, noting that the
dimension was always bigger in this case than the naive expectation. We
generalize their results to treat a much broader range of fixed-determinant
Brill-Noether loci. The main technique is a careful study of symplectic
Grassmannians and related concepts.
