  We use intermediate symplectic characters to give a proof and variations of
Hopkins' conjecture, now proved by Hopkins and Lai, on the number of shifted
plane partitions of shifted double staircase shape with bounded entries. In
fact, we prove some character identities involving intermediate symplectic
characters, and find generating functions for such shifted plane partitions.
The key ingredients of the proof are a bialternant formula for intermediate
symplectic characters, which interpolates between those for Schur functions and
symplectic characters, and the Ishikawa-Wakayama minor-summation formula.
