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 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. 1 However, these compactifications are not related to the toroidal compactifications of Shimura varieties but, rather, to the "reductive Borel-Serre" compactification.

Beyond Endoscopy for the Relative Trace Formula Ii: Global Theory

“…This is necessary, even in the simple cases that we are considering here, in order to include "pure inner forms" of the spaces under consideration into the picture and get a complete comparison between relative trace formulas. The appropriate notions for harmonic analysis on stacks were developed in [Sak16]; however in this paper we will only use the notion of stacks symbolically, and explicitly define the spaces of test functions that we need, without making use of that theory.…”

“…The appearance of pure inner forms in the conjectures can also be understood in terms of the relative trace formula, and more precisely in terms of the quotient stack (X × X)/G, cf. [SV, §16.5], [Sak16]. 2 We are really referring to stable or "quasi-stable" trace formulas here, e.g., in the case of the Arthur-Selberg trace formula the individual summands of the invariant trace formula which are matched with stable trace formulas of endoscopic groups; these summands can be considered as "quasi-stable" trace formulas with their own L-group, namely the corresponding endoscopic L-group.…”

Relative Functoriality and Functional Equations via Trace Formulas

“…For example, for X = T \ PGL2, X α includes of (1.3) will be denoted by S(X × X/G) or S(X), by abuse of notation. (This notation was used in [Sak16] for the space S(X × X) G of G-coinvariants; in our examples, the push-forwards of measures to C X will correspond to stable coinvariants. These are the "test measures" for stable trace formulas.…”

“…The geometric decomposition of the relative trace formula or, at least, of its "stable" part (which is often the whole RTF), is expressed in terms of the spaces S(X(k v )) of "stable orbital integrals" of the various completions of k. I point the reader to [Sak16,§6.4] for an attempt at a general formulation, where the relative trace formula is presented as a sum of the form:…”

“…Here, again, the natural setting to define automorphic spectra is that of a generalization of the trace formula, namely the relative trace formula of Jacquet. In modern language, the relative trace formula is a distribution on the adelic points of a quotient stack X = [(X × X ′ )/G diag ] [Sak16], where X, X ′ are two spherical varieties (possibly the same) for the same group G, together with two ways to write this distribution, one "geometric" and one "spectral". The content of the spectral side is, by definition, the "automorphic spectrum" of the pair (X, X ′ ) (or of the space X, if X ′ = X).…”