Composition series for analytic continuations of holomorphic discrete series representations of (
Composition series for analytic continuations of holomorphic discrete series representations of (
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(的全纯离散级数表示的解析延拓的组合级数
DOI：
10.1090/s0002-9947-1980-0574799-x
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发表时间：
1980
影响因子：
1.3
通讯作者：
B. Ørsted
中科院分区：
文献类型：
--
作者：
B. Ørsted
研究了半单李群G = SU（n，n）的一类全纯离散级数表示及其在诱导参数X中的解析延拓.在X的值的表示成为可约的，我们计算的复合系列的条款的Peter-Weyl基的Herinitian对称空间G的Shilov边界。导论. Hermitian对称空间G/' K上的全纯向量丛E已被广泛地研究，特别是与G的全纯离散级数的实现有关。我们知道，在某些情况下，通过考虑G的Harish-Chandra模M由E的AT-有限部分组成，有可能在定义E的K表示的参数中获得解析延拓，从而产生G的新的（非离散）酉表示[5]。这些是M的底子项，由Shilov边界上的微分方程定义。本文将描述群SU（n，n）的线丛的特殊情况下M的全合成级数及其泛覆盖。我们对复合级数的计算不是基于（像往常一样）交织算子（部分原因是子项似乎是新的，大部分是非幺正的表示），而是基于M上的G-不变（一般是不定的）埃尔米特形式的行为。基本公式是U（n）上分布det（1- x）~x在Peter-Weyl L ~ 2基上的展开。在§1中，我们推导出这个“二项式公式”，给出了M上不变的埃尔米特形式的确定，在§2中，我们推导出用U（n）的全纯对偶的某种序表示的复合级数。最后在§3中给出了模的Ti-谱和A/-谱的一个结果和一个猜想。作者从与H.雅各布森岛E.西格尔，B。Speh（他的论文[8]对51/（2，2）以及SO（n，2）的情形作了彻底的处理）和M.韦尔涅1.设G表示群SU（n，n）[1]，其中n是固定的，K是同构于（7（1）XSU（n）XSU（n）的极大紧子群.字符X*：e»^eTM（1）1979年10月9日由编辑接收。AMS（MOS）主题分类（1970年）。小学22 E45、43 A85;中学05 A10、32 H10。
We study a certain family of holomorphic discrete series representations of the semisimple Lie group G = SU(n, n) and the corresponding analytic continuation in the inducing parameter X. At the values of X where the representations become reducible, we compute the composition series in terms of a Peter-Weyl basis on the Shilov boundary of the Herinitian symmetric space for G. Introduction. Holomorphic vector bundles E over a Hermitian symmetric space G/' K have been studied extensively, in particular in connection with realization of the holomorphic discrete series for G. One knows in certain cases that by considering the Harish-Chandra module M for G consisting of AT-finite sections of E, it is possible to obtain an analytic continuation in the parameters of the representation of K defining E, so that new (nondiscrete) unitary representations of G ensue [5]. These are bottom subquotients of M and defined by differential equations on the Shilov boundary. In this paper we will describe the full composition series for M in the special case of line bundles for the group SU(n, n) and its universal covering. Our computation of the composition series is not based (as usual) on intertwining operators (partly because the subquotients seem to be new, mostly nonunitary, representations) but rather on the behavior of the G-invariant (nondefinite in general) Hermitian form on M. The basic formula is an expansion of the distribution det(l — x)~x on U(n) in terms of the Peter-Weyl L2-basis. In §1 we derive this "binominal formula" affording a determination of the invariant Hermitian form on M, and in §2 we derive the composition series, expressed in terms of a certain ordering of the holomorphic dual of U(n). Finally in §3 we list a result and a conjecture relating the Ti-spectrum and the A/-spectrum of the modules. The author has benefited from conversations with H. P. Jakobsen, I. E. Segal, B. Speh (whose paper [8] gives a thorough treatment of the case of 51/(2, 2) as well as SO(n, 2)) and M. Vergne. 1. Let G denote the group SU(n, n) [1] for a fixed n and K its maximal compact subgroup which is isomorphic to (7(1) X SU(n) X SU(n). The character X*:e»^eTM (1) Received by the editors October 9, 1979. AMS (MOS) subject classifications (1970). Primary 22E45, 43A85; Secondary 05A10, 32H10.