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The Schwartz space of a smooth semi-algebraic stack

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A Correction to this article was published on 03 October 2018

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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.

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Change history

  • 03 October 2018

    The purpose of this note is to fix two gaps in the construction of Schwartz spaces of semi-algebraic stacks in [4], and to strengthen some statements, replacing quasi-isomorphisms by homotopy equivalences.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Rutgers University – Newark, 101 Warren Street, Smith Hall 216, Newark, NJ, 07102, USA

    Yiannis Sakellaridis

  2. Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Heroon Polytechneiou 9, 15780, Zografou, Greece

    Yiannis Sakellaridis

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Correspondence to Yiannis Sakellaridis.

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To Joseph Bernstein, in admiration and gratitude

This work was supported by NSF grants DMS-1101471 and DMS-1502270.

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Sakellaridis, Y. The Schwartz space of a smooth semi-algebraic stack. Sel. Math. New Ser. 22, 2401–2490 (2016). https://doi.org/10.1007/s00029-016-0285-3

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