一类球面簇的调和分析
Harmonic analysis on certain spherical varieties
摘要：
布雷弗曼和卡日丹提出了一个猜想，后来由吴廷琰加以改进，并由萨凯利亚尔扩展到球面簇的框架，该猜想断言仿射球面簇允许施瓦兹空间、傅里叶变换和泊松求和公式。第一作者与B.~刘和后来的前两个作者证明了这些结构的某些球形品种$Y$建成的三元组的二次空间。然而，在这些作品中，傅里叶变换只被证明存在。本文首次给出了$Y.$上傅里叶变换的一个显式公式，并证明了它在非阿基米德情形下是酉的.作为这一结果的预备，我们给出了可裂单连通群的极大抛物子群上的Braverman-Kazhdan空间的傅里叶变换的显式公式.这些傅里叶变换是独立的利益，例如，从解析数论的观点。
英文摘要：
Braverman and Kazhdan proposed a conjecture, later refined by Ngô and broadened to the framework of spherical varieties by Sakellaridis, that asserts that affine spherical varieties admit Schwartz spaces, Fourier transforms, and Poisson summation formulae. The first author in joint work with B.~Liu and later the first two authors proved these conjectures for certain spherical varieties $Y$ built out of triples of quadratic spaces. However, in these works the Fourier transform was only proven to exist. In the present paper we give, for the first time, an explicit formula for the Fourier transform on $Y.$ We also prove that it is unitary in the nonarchimedean case. As preparation for this result, we give explicit formulae for Fourier transforms on Braverman-Kazhdan spaces attached to maximal parabolic subgroups of split, simple, simply connected groups. These Fourier transforms are of independent interest, for example, from the point of view of analytic number theory.
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