tailieunhanh - Đề tài " Hilbert series, Howe duality and branching for classical groups "

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. | Annals of Mathematics Hilbert series Howe duality and branching for classical groups By Thomas J. Enright and Jeb F. Willenbring Annals of Mathematics 159 2004 337 375 Hilbert series Howe duality and branching for classical groups By Thomas J. Enright and Jeb F. Willenbring 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 HL q of the form HL q 1 _ q GKdimL 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 - 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. 1. Introduction Let V be a complex vector space of dimension n with a nondegenerate symmetric or skew symmetric form. Let G be the group leaving the form invariant. Now G is either the orthogonal group O n or the sympletic group Sp n for n even. The representations Fx of Gl V are parametrized by the partitions A with at most n parts. In 1940 D. E. Littlewood gave a formula for the decomposition of Fx as a representation of G by restriction. The second author has been supported by the Clay Mathematics Institute Liftoff Program. 338 THOMAS J. ENRIGHT AND JEB F. WILLENBRING Theorem 1 Littlewood Restriction Lit 1 2 . Suppose that X is a partition having at