ON HARISH-CHANDRA'S GENERALIZED C-FUNCTIONS.
ON HARISH-CHANDRA'S GENERALIZED C-FUNCTIONS.
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关于 HARISH-Chandra 的广义 C 函数。
DOI：
10.2307/2373718
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发表时间：
1975
影响因子：
1.7
通讯作者：
N. Wallach
中科院分区：
文献类型：
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作者：
N. Wallach
1. Initroduction。1990年，Harish-Chandra推测他的“广义c函数”是基本的。也就是说，它们的矩阵项是由（仿射）根组成的p函数中的有理函数（见第7节的相关定义和定理7.1的精确表述）。在Schiffman[6]中，证明了该猜想对于与最小抛物相关的c函数是正确的（当半单李群是线性的情况下）。在Wallach[7]第8.11节中，给出了一种证明Schiffman结果的方法，这种方法很容易推广到具有有限中心的半简单李群的情况。理论上，Wallach[7]技术也允许计算这些c函数。本文通过简化到最小抛物情形，证明了上述猜想对线性半简单李群是成立的。根据上面的评论（以及对本文的粗略阅读），人们看到线性的假设只是为了技术上的方便。然而，这一猜想的动机来自于对一个全局领域的约化代数群的研究。Harish-Chandra已经证明了与所有有限位相关的c函数都是初等的。该结果与本文的结果相结合，表明全局c函数是一个“欧拉积”。本文不仅证明了Harish-Chandra猜想。定理6.1指出半简单李群的抛物诱导表示的缠结积分的傅里叶系数可以用p函数表示。给出了交织积分的极点结构，并给出了亚纯延拓的存在性。极点（以及零点）位于离散集合的事实简化了对互补级数的搜索。它说，如果X是酉的，有5ew (A)所以
1. Initroduction. In [4], Harish-Chandra conjectured that his "generalized C-functions" are efementary. That is, that they have matrix entries that are rational functions in P-functions composed with (affine) roots (see Section 7 for the pertinent definitions and Theorem 7.1 for the precise statement). In Schiffman [6] it is shown that the conjecture is correct for C-functions associated with minimal parabolics (in the case when the semi-simple Lie group is linear). In Wallach [7] Section 8.11, a technique for proving the result of Schiffman is given which is easily seen to generalize to the case of semi-simple Lie groups with finite center. The technique of Wallach [7] also in theory allows one to compute these C-functions. In this paper we prove that the above mentioned conjecture is true for linear semi-simple Lie groups, by reducing it to the minimal parabolic case. In light of the remarks above (and a cursory reading of this paper) one sees that the assumption of linearity is only for technical convenience. However, the motivation for this conjecture comes from the study of reductive algebraic groups over a global field. Harish-Chandra has shown that the C-functions associated to all finite places are elementary. That result combined with the results in this paper suggest that the global C-function is an "Euler product." We prove more than the conjecture of Harish-Chandra in this paper. Theorem 6.1 says that the Fourier coefficients of intertwining integrals of parabolic induced representatives of semi-simple Lie groups are expressible in terms of P-functions. It also gives the pole structure of the intertwining integrals and implies the existance of meromorphic continuation. The fact that the poles (and hence the zeros) lie in a discrete set simplifies the search for complimentary series. (It says that if X is unitary and there is s E W(A) so that