Spherical functions of the principal series representations of Sp (2, R ) as hypergeometric functions of C 2 -type
Spherical functions of the principal series representations of Sp (2, R ) as hypergeometric functions of C 2 -type
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作为 C 2 型超几何函数的 Sp (2, R ) 主级数表示的球函数
DOI：
10.2977/prims/1195162717
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发表时间：
1997
影响因子：
1.2
通讯作者：
M. Iida
中科院分区：
文献类型：
--
作者：
M. Iida
本文明确地确定了Sp（2，R）的主级数和广义主级数表示为非平凡K-typQs的球函数所满足的微分方程组。然后得到了广义主级数表示的球函数的级数展开式和积分公式。我们将定义球面函数.设G是一个真实的约化李群，K是G的极大紧子群，Po= MoA 0 No是G的一个抛物子群.设Hn是G的一个容许表示，（r，Vr），（#，Vri）是包含在HK中的K的不可约表示.我们把Horru（Vr，C7（K\G））的元素称为^ C？（K\G）®KV？= C~，r（K\G/K）-（#，r）型球函数，其中CTM（K\G）是K\G上与V？而V？是VT的矛盾表示。令θ E：Horri（ai K}（Hn，CTM（K\G}}）且i = Horrid Vr，HTC\，则θ i是附接到H* 的球函数。关于一维K型球函数所满足的微分方程组，已有许多研究。此外，它们被推广为具有连续参数的Weyl群不变交换微分算子，这是通过推广根重数而引入的（cf. [DGl]、[DG2]、[HI]、[HO]、[Ko]、[OO]、[Opl]、[Op2]、[Os]、[OS]、[Sh]）。另一方面，对向量值球函数的研究却很少。此外，球函数除了秩为1的情况外，很少以显式形式计算。因此，明确研究高阶李群上的向量值球函数是一个有趣的问题。
In this article we determine explicitly the systems of differential equations satisfied by spherical functions with non-trivial K-typQs of the principal series and the generalized principal series representations of Sp(2, R). Then we obtain series expansions and integral formulas of spherical functions of the generalized principal series representation. We shall define spherical functions. Let G be a real reductive Lie group and K be maximal compact subgroup of G, Po=MoA0No be a parabolic subgroup of G. Let Hn be an admissible representation of G and (r, Vr), (#, Vri) be irreducible representations of K which is contained in HK. We call elements of Horru( Vr, C7(K\G))^ C?(K\G)®KV? = C~,r(K\G/K) spherical functions of type-(#, r), where CTM(K\G) is the space of smooth sections of the homogeneous vector bundle over K\G associated to V? and V? is the contragredient representation of VT. Let 0E:Horri(ai K}(Hn, CTM(K\G}} and i^ Horrid Vr, HTC\ then °i is a spherical function attached to H*. There are many studies on the system of differential equations satisfied by spherical functions for 1-dimensional K-typzs. Moreover they are generalized as the Weyl group invariant commuting differential operators with continuous parameters, which are introduced by generalizing root multiplicities (cf. [DGl], [DG2], [HI], [HO], [Ko], [OO], [Opl], [Op2], [Os], [OS], [Sh]). On the other hand, there are few studies for vector-valued spherical functions. Besides, spherical functions are rarely calculated in explicit forms except for rank one cases. Therefore it is interesting to study vector-valued spherical functions of higher rank Lie group explicitly.