We consider the problem of calculating explicitly the restriction to a subgroup of some unitary representations of classical non-compact orthogonal, unitary, and symplectic groups. The representations correspond to small coadjoint orbits, namely the ones that are either elliptic or nilpotent and of minimal dimension, and the subgroups are certain dual pairs. Our method relies on conformal construction of representations. We also obtain a limit relation between the elliptic and the nilpotent case, both at the level of orbits and representations. In the nilpotent case the results are analogous to the ones for Howe dual pairs, in particular we find an explicit correspondence between subsets of the unitary duals.
© Toshiyuki Kobayashi