Some properties of square-integrable representations of semisimple Lie groups
Some properties of square-integrable representations of semisimple Lie groups
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半单李群的平方可积表示的一些性质
DOI：
10.2307/1971043
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发表时间：
1975
影响因子：
4.9
通讯作者：
W. Schmid
中科院分区：
文献类型：
--
作者：
W. Schmid
在紧李群的不可约表示理论中，权的重数公式和所谓的最高权定理是最重要的结果之一。至少在猜想上，这两种说法都与半单李群的离散表示序列有类似之处。设G是连通的半单李群，KcG是极大紧子群，且设RkK=RkG，则G有一个非空的离散级数[8]。布拉特纳猜想预测了给定的离散级数表示如何在K的作用下分解；精确的表述可以在[10]、[15]、[16]中找到。形式上，猜想的重数公式看起来就像一个权重的重数公式。关于该猜想的部分结果已在[10]、[15]中得到证明。最近，对于那些线性群G，其商G/K允许厄米对称结构[16]，建立了完整的猜想。随着这篇论文的完成，H.Hecht和我通过推广[16]的论点，成功地证明了所有线性群的Blattner猜想。根据Blattner猜想，任何特定的离散级数表示7t都包含一个独特的重数的不可约K-模V；此外，w不包含具有最高权的不可约K-模，在适当的意义下，它比V的权小。对于“大多数”离散级数表示，已知这两个性质在G[10]、[15]的所有不可约表示中具有直到无穷小的等价性。在这篇文章中，我将给出所有离散级数表示的最低K型无穷小刻画。其结果如下面的定理(1.3)所述，非常类似于最高权的定理。我还将从中得出一些结论。这篇论文的方法有一些进一步的、不那么直接的影响，这些结果将在其他地方采用。在引言的其余部分，我假设G是一个线性群。
In the theory of irreducible representations of a compact Lie group, the formula for the multiplicity of a weight and the so-called theorem of the highest weight are among the most important results. At least conjecturally, both of these statements have analogues for the discrete series of representations of a semisimple Lie group. Let G be a connected, semisimple Lie group, K c G a maximal compact subgroup, and suppose that rk K = rk G. Exactly in this situation, G has a non-empty discrete series [8]. Blattner's conjecture predicts how a given discrete series representation should break up under the action of K; precise statements can be found in [10], [15], [16]. Formally, the conjectured multiplicity formula looks just like the formula for the multiplicity of a weight. Partial results toward the conjecture have been proved in [10], [15]. More recently, the full conjecture was established for those linear groups G, whose quotient G/K admits a Hermitian symmetric structure [16]. As this paper was being completed, H. Hecht and I succeeded in proving Blattner's conjecture for all linear groups, by extending the arguments of [16]. According to Blattner's conjecture, any particular discrete series representation 7t contains a distinguished irreducible K-module V, with multiplicity one; moreover, w contains no irreducible K-module with a highest weight which is lower, in the appropriate sense, than that of V,:. For "most" discrete series representations, it was known that these two properties characterize at, up to infinitesimal equivalence, among all irreducible representations of G [10], [15]. In this paper, I shall give an infinitesimal characterization, by lowest K-type, for all discrete series representations. The result, which is stated as Theorem (1.3) below, closely resembles the theorem of the highest weight. I shall also draw a number of conclusions from it. The methods of this paper have some further, less immediate consequences, which will be taken up elsewhere. For the remainder of the introduction, I assume that G is a linear group.
某些p基团诱导特征的不可约性
DOI：
--
发表时间：
2004
期刊：
Transactions of Kokushikan University Faculty Engineering 37
影响因子：
--
作者：
中島 晴久;中島 晴久;石橋 宏行;関口 勝右;Nakajima Haruhisa;Ishibashi Hiroyuki;Sekiguchi Katsusuke
通讯作者：
Sekiguchi Katsusuke