On integrable representations of a semisimple lie group
On integrable representations of a semisimple lie group
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半单李群的可积表示
DOI：
10.1007/bf01351699
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发表时间：
1976
影响因子：
1.4
通讯作者：
W. Schmid
中科院分区：
文献类型：
--
作者：
Henryk Hecht;W. Schmid
设 G 为连通的半单李群，KCG 为最大紧子群。如果 G 的至少一个矩阵系数（或者等价地，每个 K 有限矩阵系数）位于 L~(G) 中，则 G 的不可约酉表示称为可积的。由于酉表示的矩阵系数是有界的，因此这些表示形成平方可积、不可约、酉表示集合的子集。 Trombi 和 Varadarajan [-6] 导出了给定平方可积表示的可积性的必要条件。在本文中，我们将证明它们的条件也是充分的，至少对于每个矩阵群 G 而言。我们的论点类似于 Enright [1] 之前使用的论点，用于处理 G= SU (n, 1) 的特殊情况。根据 Harish-Chandra [2] 的准则，当且仅当 K 与 G 具有相同的秩时，G 才允许平方可积表示。我们假设情况是这样，并且我们选择 G 的紧嘉丹子群 H，具有 HCK。我们将遵循[3, 5]的符号。因此，go、f0、bo 是 G、K、H 的李代数，g、f、b 是它们的复化。 (g,[~) 的根系 ~ b 是 q~" 和 4~" 的不相交并集，分别是 q~ 中的紧根和非紧根的集合。通过求幂，H 的字符组 H 变得与晶格 AC ib* 同构。 (g，b)和(f，b)的Weyl基团将被称为| 4∨和~K 为了能够使用[3, 5]的结果，我们假设G是一个矩阵群。正如[3, 5]中所指出的，可以通过进一步的工作来消除这个假设。一旦 G 被假设为线性，为了符号简单性，并且不失一般性，我们可以强加条件 G 具有简单连接的复数。 (1) Harish-Chandra [2] 已对平方可积、不可约、酉表示的同构类集合进行了参数化：对于每个非奇异 1 2cA，都有一个对应的表示 rcz，反之亦然； rex 和 rt，当 w2=# 时，精确地同构，对于某些我们 W。Trombi-Varadarajan 的结果断言，如果 rc~ 可积，则
Let G be a connected, semisimple Lie group, and KCG a maximal compact subgroup. An irreducible unitary representation of G is called integrable if at least one of its matrix coefficients-or equivalently, every K-finite matrix coefficientlies in L~(G). Since the matrix coefficients of a unitary representation are bounded, these representations form a subset of the set of square-integrable, irreducible, unitary representations. Trombi and Varadarajan [-6] have derived a necessary condition for the integrability of a given square-integrable representation. In this note, we shall show that their condition is also sufficient, at least for every matrix group G. Our arguments resemble those previously used by Enright [1], to treat the special case of G= SU (n, 1).According to a criterion of Harish-Chandra [2], G admits square-integrable representations if and only if K has the same rank as G. We assume that this is the case, and we choose a compact Cartan subgroup H of G, with HCK. We shall follow the notation of [3, 5]. Thus go, f0, bo are the Lie algebras of G, K, H, and g, f, b their complexifications. The root system~ b of (g,[~) is the disjoint union of q~" and 4~", the sets of, respectively, compact and noncompact roots in q~. Via exponentiation, the character group H of H becomes isomorphic to a lattice AC ib*. The Weyl groups of (g, b) and (f, b) wilt be referred to as| 4¢~ and~ K In order to be able to use the results of [3, 5], we suppose that G is a matrix group. As pointed out in [3, 5], this assumption can be removed with some further work. Once G is assumed linear, for notational simplicity, and without loss of generality, we may impose the condition G has a simply connected complexification.(1) The set of isomorphism classes of square-integrable, irreducible, unitary representations has been parameterized by Harish-Chandra [2]: to each nonsingular 1 2cA, there corresponds such a representation rcz, and conversely; rex and rt, are isomorphic precisely when w2=#, for some we W. A result of Trombi-Varadarajan asserts that if rc~ is integrable, then