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广义Harish-Chandra下降、Gelfand对和Jacquet-Rallis定理的阿基米德类比

Generalized Harish-Chandra descent, Gelfand pairs and an Archimedean analog of Jacquet-Rallis' Theorem

The Weizmann Institute of Science（魏茨曼科学研究所）

Avraham Aizenbud, Dmitry Gourevitch, Eitan Sayag

备注：
A merge of arXiv:0803.3395 and arXiv:0803.3397 (with no additional material). Appendix D by Avraham Aizenbud, Dmitry Gourevitch and Eitan Sayag. v2,v3: minor changes. v4: correction in the speciality criterion (7.3.7). v5: minor correction in the proof of Proposition 7.2.1. v6: corrections, including a change in the defintion of adapted distribution
发表信息：
Duke Math. J. 149, no. 3 (2009), 509-567
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摘要：

在本文的第一部分中，我们推广了下降技术由于Harish-Chandra的情况下，约化群作用于光滑仿射簇都定义在任意局部域F的特征为零。我们的主要工具是月球切片定理。在本文的第二部分中，我们将这种技术应用于对称对。特别地，我们证明了对（GL（n+k，F），GL（n，F）× GL（k，F））和（GL（n，E），GL（n，F））对任何局部域F及其二次扩张E都是Gelfand对.在非阿基米德的情况下，第一个结果已经被Jacquet和Rallis证明，第二个结果已经被Flicker证明。我们还证明了GL（n，F）上的任何共轭不变分布都是关于转置不变的。对于非阿基米德F，后者是Gelfand和Kazhdan的经典定理。

英文摘要：

In the first part of the paper we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna Slice Theorem. In the second part of the paper we apply this technique to symmetric pairs. In particular we prove that the pairs (GL(n+k,F), GL(n,F) x GL(k,F)) and (GL(n,E), GL(n,F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proven earlier by Jacquet and Rallis and the second by Flicker. We also prove that any conjugation invariant distribution on GL(n,F) is invariant with respect to transposition. For non-Archimedean F the latter is a classical theorem of Gelfand and Kazhdan.

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