The Schwartz space of a smooth semi-algebraic stack

TL;DR

This paper extends the concept of Schwartz functions to smooth semi-algebraic stacks, providing a geometric framework that generalizes trace formulas and offers insights into Langlands conjectures.

Contribution

It introduces a new definition of Schwartz functions on semi-algebraic stacks and constructs evaluation maps at semisimple points, linking geometric stacks to trace formulas.

Findings

Defined Schwartz functions on semi-algebraic stacks.

Constructed evaluation maps at semisimple points.

Connected stack theory with trace formulas and Langlands conjectures.

Abstract

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 global 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.