Regularity properties of generalized Harish-Chandra expansions
Regularity properties of generalized Harish-Chandra expansions
复制标题
广义 Harish-Chandra 展开式的正则性质
DOI：
10.4064/bc55-0-17
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
A. Pasquale
中科院分区：
文献类型：
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作者：
G. Ólafsson;A. Pasquale
研究了用广义Harish-Chandra展开式表示的正Weyl腔A上函数的正则性。我们采用了Heckman和Opam的方法，允许任意Weyl群不变复数。我们所考虑的广义Harish-Chandra展开式与根的任意抛物系统Θ和根的多重数函数有关。它们是广义超几何函数的Θ的Weyl群上的和。它们在A上是解析的，在谱参数λ中是亚纯的。我们证明了它们作为WΘ不变全纯函数在(WΘ·A+)的管状邻域上扩张。λ奇点的可能位置被证明是由固定数据自然构造的显式函数的极集。对于具有偶数重数的约化根系统，我们改进了我们的结果，并证明了λ奇点位于仿射超平面的特定有限族上。最后，当所有重数都等于2时，我们推广了具有复杂结构的黎曼对称空间上球函数的经典显式公式。导言。本文研究了用广义Harish-Chandra展开式表示的某些函数的正则性。这些函数有两个重要的特例：(1)非紧型黎曼对称空间上的Harish-Chandra球函数；(2)非紧因果对称空间上的球函数。2000年数学学科分类：43A90、33C67。
We study the regularity properties of functions that can be represented on a positive Weyl chamber A by generalized Harish-Chandra expansions. We adopt the approach of Heckman and Opdam by allowing arbitrary Weyl-group-invariant complex multiplicities. The generalized Harish-Chandra expansions that we consider are associated with arbitrary parabolic system Θ of roots and a root multiplicity function. They are given as sum over the Weyl group of Θ of generalized hypergeometric functions. They are analytic on A and meromorphic in the spectral parameter λ. We prove that they extend as WΘ-invariant holomorphic functions on a tubular neighborhood of (WΘ · A+). The possible location of the λ-singularities is shown to be the polar set of an explicit function naturally constructed from the fixed data. For reduced root systems with even multiplicities we refine our result and show that the λ-singularities lie on a specific finite family of affine hyperplanes. Finally, when all multiplicities are equal to 2, we generalize the classical explicit formula for spherical functions on Riemannian symmetric spaces with a complex structure. Introduction. In this paper we study the regularity properties of certain functions which are represented by generalized Harish-Chandra expansions. There are two important special instances of these functions: (1) Harish-Chandra’s spherical functions on Riemannian symmetric spaces of the noncompact type; (2) spherical functions on noncompactly causal symmetric spaces. 2000 Mathematics Subject Classification: 43A90, 33C67.