GL（n，R）和GL（n，C）光滑表示的导数
Derivatives for smooth representations of GL(n,R) and GL(n,C)
摘要：
在p-adic情形下GL（n）的光滑表示的导数的概念由J. Bernstein和A.泽列文斯基在阿基米德的情况下，最高导数的类似物被定义为不可约的酉表示由S。Sahi和所谓的“引证”表示。在本文中，我们定义的衍生物的所有顺序光滑容许弗雷歇表示（适度增长）。阿基米德情形比p-adic情形更有问题;例如任意导数不需要是可容许的。然而，最高导数仍然是允许的，并且对于不可约的unitarizable表示，与所引表示的光滑向量空间一致。在[AGS]中，我们证明了最高导函子的正合性，并计算了所有单项表示的最高导。证明了最高导函子的正合性，并计算了所有单项表示的最高导。我们应用这些结果完成了所有不可约酉表示的导出表示的计算，并证明了退化Whittaker模型对于酉表示的唯一性，从而完善了[Sah 89，Sah 90，SaSt 90，GS 12]的结果.
英文摘要：
The notion of derivatives for smooth representations of GL(n) in the p-adic case was defined by J. Bernstein and A. Zelevinsky. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations by S. Sahi and called the "adduced" representation. In this paper we define derivatives of all order for smooth admissible Frechet representations (of moderate growth). The archimedean case is more problematic than the p-adic case; for example arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation. In [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations. We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS12].
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