# 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.

## Authors

Thomas J. Enright

Department of Mathematics, University of California at San Diego, La Jolla, CA 92093-0112, United States

Jeb F. Willenbring

Department of Mathematics, University of Wisconsin-Milwaukee, Milwaukee, WI 53201-0413, United States