Orbital Integrals in Reductive Groups
Orbital Integrals in Reductive Groups
复制标题
约简群中的轨道积分
DOI：
10.2307/1970822
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发表时间：
1972
影响因子：
4.9
通讯作者：
R. Rao
中科院分区：
文献类型：
--
作者：
R. Rao
设G是定义在特征为零的（非离散）局部紧域k上的连通约化线性代数群，G是它的k-有理点群.若O（x）= {xy '：yeG}是x的轨道，则已知O（x）是局部紧的（在G的Hausdorff拓扑中）且同胚于G/G，.已知各向同性子群Gx是幺模的（[2]，p.235），因此空间G/GX带有G-不变Radon测度dy*（y* =yGx）。一个在调和分析中似乎有些重要的问题是，移植到轨道O（x）上的这个测度是否是G中的Radon测度，即，对于G的紧子集是有限的。如果是这样，那么积分
Let G be a connected reductive linear algebraic group defined over a (nondiscrete) locally compact field k of characteristic zero, and G be its group of k-rational points. If O(x) = {yxy': y e G} is the orbit of x, then it is known2 that O(x) is locally compact (in the Hausdorff topology of G) and homeomorphic to G/G,. The isotropy subgroup Gx is known to be unimodular ([2], p. 235) and so the space G/GX carries a G-invariant Radon measure dy*(y* =yGx). A question that seems to be of some importance in harmonic analysis is, whether this measure, transplanted to the orbit O(x), is a Radon measure in G, i.e., is finite for compact subsets of G. If this is the case then the integral