The Schwartz space of a smooth semi-algebraic stack

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.

96pp. Erratum added at the end to fix two gaps and strengthen some statements from quasi-isomorphism to homotopy equivalence

References in corpus (4)

• Symplectic local root numbers, central critical L-values, and restriction problems in the representation theory of classical groups
• Schwartz functions on Nash manifolds
• Foundations of Topological Stacks I
• The local theta correspondence and the local Gan-Gross-Prasad conjecture for the symplectic-metaplectic case

Cited by in corpus (4)

• Functorial transfer between relative trace formulas in rank one
• Relative de Rham Theory on Nash Manifolds
• Moduli spaces and algebraic cycles in real algebraic geometry
• On Schwartz equivalence of quasidiscs and other planar domains