On certain small representations of indefinite orthogonal groups
On certain small representations of indefinite orthogonal groups
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关于不定正交群的某些小表示
DOI：
10.1090/s1088-4165-97-00031-9
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发表时间：
1997
期刊：
影响因子：
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通讯作者：
Jing Huang
中科院分区：
文献类型：
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作者：
Chengru Zhu;Jing Huang
对于任意n∈n，使得2n≤min(p, q)，我们构造了一个πn （O(p, q)）的表示πn (p, q)，其中p+q为交换集的核，该交换集由n(n+1) 2个O（p, q）不变微分算子组成，在各向同性锥上具有显著GLn(R)齐次度。通过用对应的形式构造的某种表示来标识πn，我们证明（除非p = q = 2n） πn的K个有限向量的空间是具有gelfund - kirillov维数n（p + q−2n−1）的O（p, q）的不可约酉表示的（g， K）模。我们的构造推广了Binegar和Zierau （SOe(p, q)， common的奇异表示的统一）的工作。数学。物理学报，138(1991),245-258)。1. 表示πn的构造设V = R ' R⊕R为具有二次型的实向量空间
For any n ∈ N such that 2n ≤ min(p, q), we construct a representation πn of O(p, q) with p+q even as the kernel of a commuting set of n(n+1) 2 number of O(p, q)-invariant differential operators in the space of C∞ functions on an isotropic cone with a distinguished GLn(R)-homogeneity degree. By identifying πn with a certain representation constructed via the formalism of the theta correspondence, we show (except when p = q = 2n) that the space of K-finite vectors of πn is the (g, K)-module of an irreducible unitary representation of O(p, q) with Gelfand-Kirillov dimension n(p + q − 2n − 1). Our construction generalizes the work of Binegar and Zierau (Unitarization of a singular representation of SOe(p, q), Commun. Math. Phys. 138 (1991), 245–258) for n = 1. 1. Construction of the representation πn Let V = R ' R ⊕ R be the real vector space equipped with the quadratic form