An extension of the Littlewood Restriction Rule is given that covers all pertinent parameters and simplifies to the original under Littlewood’s hypotheses. Two formulas are derived for the Gelfand-Kirillov dimension of any unitary highest weight representation occurring in a dual pair setting, one in terms of the dual pair index and the other in terms of the highest weight. For a fixed dual pair setting, all the irreducible highest weight representations which occur have the same Gelfand-Kirillov dimension.
We define a class of unitary highest weight representations and show that each of these representations, L, has a Hilbert series HL(q) of the form:
where R(q) is an explictly given multiple of the Hilbert series of a finite dimensional representation B of a real Lie algebra associated to L. Under this correspondence L→B , the two components of the Weil representation of the symplectic group correspond to the two spin representations of an orthogonal group. The article includes many other cases of this correspondence.