Hilbert series, Howe duality and branching for classical groups. (English) Zbl 1087.22011

Enright, Thomas J.; Willenbring, Jeb F.

Ann. Math. (2) 159, No. 1, 337-375 (2004).

The authors consider an extension of the Littlewood restriction rule which covers all pertinent parameters and reduces to the original one under Littlewood’s hypothesis. Two formulas are derived for the Gelfand-Kirillov dimension of any unitary highest weight representation occuring in the dual pair setting, one in terms of the dual pair index and the other in terms of the highest weight. They define a class of unitary highest weight representations and show that each of these representations has a Hilbert series.
In Section 1 they present the main results of this article, most of which were announced in [T. J. Enright and J. F. Willenbring, Proc. Natl. Acad. Sci. USA 100, No. 2, 434–437 (2003; Zbl 1065.22008)]. In the following sections they give details of the proofs of these results and some remarks. These descriptions are too deep and complicated to explain here.

Reviewer: Takayuki Nôno (Hiroshima)

MSC:

22E45	Representations of Lie and linear algebraic groups over real fields: analytic methods
22E47	Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.)
17B10	Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)

Keywords:

representation of classical groups; unitarizable highest weight representations

Citations:

Zbl 1065.22008