Hilbert series, Howe duality and branching for classical groups

Abstract

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 ${\rm H}_L(q)$ of the form: \[{\rm H}_L(q)={1\over (1-q)^{{\rm GKdim}\, L}} R(q), \] 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 $L\rightarrow B$ , 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.