An explicit integral representation of Whittaker functions for the representations of the discrete series : The case of $SU(2,2)$
An explicit integral representation of Whittaker functions for the representations of the discrete series : The case of $SU(2,2)$
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离散级数表示的 Whittaker 函数的显式积分表示：$SU(2,2)$ 的情况
DOI：
10.1215/kjm/1250518270
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发表时间：
1997
影响因子：
--
通讯作者：
T. Oda
中科院分区：
文献类型：
--
作者：
T. Oda
从某种意义上说，本文是对Yamashita [12]的补充。它也类似于[6]中的一个结果。我们考虑L ie群G = SU（2，2）. G的广义离散级数表示是关于G的极大单位子群N的一个非退化特征标的W hittaker模型。利用Schmid定理，Y [12]对离散级数表示的W hittaker模型中极小K型向量所满足的微分方程进行了推广.本文的目的是将这一计算进一步推广到一个更一般的表达式中，即表示这些向量的W hittaker函数属于m个纯K型（定理4.4和4.5）。这是Jacquet对W hittake r函数的一般集成表示。B u表示的是跟踪表中的某些项。我们希望我们的公式对研究真实的场所上离散的自同构表示的L因子是有用的。在§ 2中，我们回顾了SU（2，2）的结构，极大紧子群K的离散性和表示. B在文献[1，2]中给出了一些符号和定义。在§3中，我们回顾了Yamashita和Kostan对W hitta k r v e c c to rs空间的定义的结果。我们还计算了模拟歌剧的平均A -p来进行实验。在§ 4中，我们描述了文[ 1 - 2 ]中的Whittaker函数完整微分方程组。进一步地，我们证明了在N的非退化特征标的宇称条件下，解析W hittaker函数的积分表示可以由微分方程得到。B由于我们对Au orphic form s的培养很长，
In a sense , th is p a p e r is a supplem ent to th e p a p e r [1 2 ] of Yamashita. A lso it is an analogue of a result in [6]. W e conside r th e L ie g roup G = S U ( 2 ,2 ) . T h e la r g e discrete series representation o f G h a s a W hittaker m odel w ith respect to a nondegenerate character o f the m ax im al un ipo ten t subgroup N o f G . U sing th e Schmid o p e ra to r , Y a m a sh ita [1 2 ] e x p lic itly c o m p u te d t h e differential equations satisfied by the minimal K type vectors in the W hittaker model of the discrete series representations. The purpose of this paper is to push this com putation one step further to o b ta in a n e x p lic it in te g ra l re p re se n ta tio n o f t h e W hittaker functions representing these vectors belonging to the m in im al K -type (Theorems 4.4 a n d 4 .5 ) . T h e r e is a genera l in tegra l representa tion d u e t o Jacquet for W hittake r func tions. B u t th is represen ta tion is som etim es in trac tab le for h igher rank g roups. W e hope ou r formula is usefu l for the investigation of L factors of autom orphic representations o f th e d isc re te se r ie s a t th e real places. T he con ten t o f th is p a p e r is a s follows: In § 2 , w e b rie fly rev iew the s tru c tu re o f S U (2 ,2 ) , th e d isc re te se r ie s a n d th e representations of the maximal compact subgroup K . B asic notations and definitions a re found in [1 ,2 ] , w h ich w e fo llow . In §3 , w e review th e re su lts o f Yamashita and of K o s ta n t o n th e d im e n s io n o f th e sp a c e o f W h itta k e r v e c to rs . W e a lso calculate th e ra d ia l A -p a rt o f th e S chm id opera to r exp lic itly . In § 4 , we describe the holonomic system of differential equations of Whittaker functions w h ic h h a s a p p e a re d i n [ 1 2 ] . F u rth e rm o re , w e sh o w t h a t t h e integral representation o f th e analytic W hittaker function can be obtained from the differential equations under the parity condition of a nondegenerate character of N. B ecause w e be long to the cu ltu re o f au tom orph ic form s, the maximal