Finite multiplicity theorems for induced representations of semisimpmle Lie groups II, -Applications to generalized Gelfand-Graev representations-
Finite multiplicity theorems for induced representations of semisimpmle Lie groups II, -Applications to generalized Gelfand-Graev representations-
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半单李群 II 的诱导表示的有限重数定理，-在广义 Gelfand-Graev 表示中的应用-
DOI：
10.1215/kjm/1250520400
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发表时间：
1988
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通讯作者：
Hiroshi Yamashita
中科院分区：
文献类型：
--
作者：
Hiroshi Yamashita
本文是我们关于诱导表示的有限重数性质工作的第二部分。本文利用第一部分[32]的结果，给出了半单李群的广义Gelfand-Graev表示的有限重数定理(下文称为S[I])。设G是中心有限的连通半单李群。在文[1]中，我们推广了van den Bann[1]关于半单对称空间的Plancerel公式中重数的有限性的结果，从而在更一般的情况下发展了球函数理论。此外，我们还给出了G的诱导表示具有有限重数性质的良好充分条件。为了统一理解S和S在不同情况下得到的诱导表示的许多有限重数定理([2]，[8]，[26]等)，我们在[2]，[8]，[26]等不同情况下得到了许多关于诱导表示的有限重数定理。在其他方面，我们感兴趣的是以下重要的考试。设G=Ka，N“，是G的岩川分解。然后N n i i S是G的极大酉子群。我们在文[1]中证明了诱导表示Ind，i(E)(可微地(=C-)或唯一地(=V)诱导)对于N的任意一维表示(=特征)e具有唯一的多重性。(见[I，4.2])。正如Wes在文献[I]的附录中所建议的那样，对这种n诱导表示的研究被归结为具有非退化特征标C的Inn m(E)的La_g e e x_te_n_t的研究。根据Shalika[26]，GGR有一个显著的性质：如果G是线性的和拟分裂的，则么正诱导的GGR具有重数1(cf.[I，定理4.5])。这些GGR不仅在表象理论本身，而且在S的自同构理论中都扮演着重要的角色。但是，在前人的表示理论中，GGR还不足以理解所有的不可约表示。换句话说，G存在许多永远不会“发生”的不可约表示。
This paper is the sccond part of our work o n finite multiplicity property for induced representations. We give in this article finite multiplicity theorems for generalized Gelfand-Graev representations of semisimple Lie groups, applying the results of the first part [32] (referred a s [I] later on). Let G be a connected semisimple Lie group with finite center. In [ I ] , we generalized th e result of van den Ban [1] on finiteness of multiplicities in the Plancherel formula associated to a semisimple symmetric space, developing the theory of spherical functions in a much more general setting. Furthermore, we gave there nice sufficient conditions for an induced representation of G to have finite multiplicity property. T h ese c r ite r io n s en ab le u s to understand, in a unified manner, many finite multiplicity theorems f o r induced representations, obtained in different situations ([2], [8], [26], etc.). Among others, we are interested in the following important exam p le . Let G=KA,N„, be an Iwasawa decomposition o f G . Then N n i i s a maximal unipotent subgroup of G . We showed in [I] that the induced representation Ind ,, i (e) (differentiably (=C -) o r un itarily (= V ) induced) has fin ite m ultip licity property f o r any one-dimensional representation (=character) e o f N in . (see [I, 4 .2 ]) . As wes suggested in Appendix of [I], the study of such a n induced representation is reduced, to a la r g e e x te n t , to th a t o f Inn m (e ) w ith a non-degenerate character C. T h e la tte r representation is called a Gelfand-Graev representation (=GGR fo r sh o rt) . According to Shalika [26], the GGRs have a remarkable property : the unitarily induced GGRs are of multiplicity one if G is linear and quasi-split (cf. [I, Theorem 4 .5 ]) . These GGRs have been playing an important role not only in the representation theory itself but also in the theory of automorphic fo rm s. (For the historical background of the study of GGRs, we refer to [31, 0.1].) But, in the representation theoretical p o in t o f v iew , GGRs a r e not large enough to understand all the irreducible representations through them. In other words, there exist numbers of irreducible representations of G that never "occur"