Let λ be a partition, with l parts, and let Fλ be the irreducible finite dimensional representation of GL(m) associated to λ when l ≤ m. The Littlewood Restriction Rule describes how Fλ decomposes when restricted to the orthogonal group O(m) or to the symplectic group Sp(m/2) under the condition that l ≤ m/2. In this paper, this result is extended to all partitions λ. Our method combines resolutions of unitary highest weight modules by generalized Verma modules with reciprocity laws from the theory of dual pairs in classical invariant theory. Corollaries include determination of the Gelfand–Kirillov dimension of any unitary highest weight representation occurring in a dual pair setting, and the determination of their Hilbert series (as a graded module for p−). Let L be a unitary highest weight representation of sp(n, R), so*(2n), or u(p, q). When the highest weight of L plus ρ satisfies a partial dominance condition called quasi-dominance, we associate to L a reductive Lie algebra gL and a graded finite dimensional representation BL of gL. The representation BL will have a Hilbert series P(q) that is a polynomial in q with positive integer coefficients. Let δ(L) = δ be the Gelfand–Kirillov dimension of L and set cL equal to the ratio of the dimensions of the zeroth levels in the gradings of L and BL. Then the Hilbert series of L may be expressed in the form In the easiest example of the correspondence L → BL, the two components of the Weil representation of the symplectic group correspond to the two spin representations of an orthogonal group.
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机译:令λ为具有l个部分的分区,令F λ sup>为当l≤m时与λ相关的GL(m)的不可约有限维表示。利特伍德约束规则描述了在l≤m / 2的条件下,F λ sup>在分解为正交群O(m)或辛群Sp(m / 2)时如何分解。在本文中,该结果扩展到所有分区λ。我们的方法结合了经典不变理论中的双对理论,将广义Verma模块的of最大权重模块的分辨率与互惠律相结合。结果包括确定在双对中发生的任何单一最高权重表示的Gelfand–Kirillov维度,以及确定其Hilbert级数(作为p - sup>的分级模块)。令L为sp(n,R),so *(2 n em>)或u( p em>, q em> )。当 L em>加上ρ的最高权重满足称为准主导的部分支配条件时,我们将 L em>的归约李代数g L em>与g L em>的分级有限维表示 BL em>。表示形式 BL em>的希尔伯特级数 P em>( q em>)是 q em>中的多项式,具有正整数系数。令δ( L em>)=δ为 L em>的Gelfand–Kirillov维度,并将 cL em>设置为等于零级维度的比率在 L em>和 BL的等级中。 em>然后, L em>的希尔伯特级数可以用以下形式表示:对应的最简单示例> L em>→ BL em>,辛群的Weil表示的两个分量对应于正交群的两个自旋表示。
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