The Schwartz space of a smooth semi-algebraic stack
The Schwartz space of a smooth semi-algebraic stack
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光滑半代数栈的 Schwartz 空间
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发表时间：
2015
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通讯作者：
Y. Sakellaridis
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作者：
Y. Sakellaridis
Schwartz函数或测度定义在任何光滑半代数(“Nash”)流形上，并且已知构成半代数受限拓扑的辫子。我们将这一定义推广到光滑半代数堆，它被定义为Nash流形范畴中的几何堆。此外，当它们是从形式为X/G的代数商堆得到的，其中X是光滑仿射簇，G是定义在数域k上的约化群，我们尽可能地在堆栈的每个半单k点上定义一个“赋值映射”，而不使用截断方法。这对应于那些轨道积分之和的正则化，这些轨道积分的半单部分对应于所选的k点。这些评估图原则上产生了一个分布，它推广了Arthur-Selberg迹公式和Jacquet的相对迹公式，尽管前者以及后者的许多实例实际上不能用本文的纯几何方法来定义。无论如何，堆栈理论的观点为朗兰德的许多版本以及相对的朗兰兹猜想中出现的纯粹的内部形式提供了一种解释。
Schwartz functions, or measures, are defined on any smooth semi-algebraic (“Nash”) manifold, and are known to form a cosheaf for the semi-algebraic restricted topology. We extend this definition to smooth semi-algebraic stacks, which are defined as geometric stacks in the category of Nash manifolds. Moreover, when those are obtained from algebraic quotient stacks of the form X/G, with X a smooth affine variety and G a reductive group defined over a number field k, we define, whenever possible, an “evaluation map” at each semisimple k-point of the stack, without using truncation methods. This corresponds to a regularization of the sum of those orbital integrals whose semisimple part corresponds to the chosen k-point. These evaluation maps produce, in principle, a distribution which generalizes the Arthur–Selberg trace formula and Jacquet’s relative trace formula, although the former, and many instances of the latter, cannot actually be defined by the purely geometric methods of this paper. In any case, the stack-theoretic point of view provides an explanation for the pure inner forms that appear in many versions of the Langlands, and relative Langlands, conjectures.