On the Schwartz space of a smooth semi-algebraic stack

There are many instances in the field of automorphic forms where nice classification results do not exist for a single group or space, but they do exist if one considers some "pure inner forms" at the same time. It has been suggested by Joseph Bernstein that this phenomenon should be explained by replacing the spaces and groups by appropriate quotient stacks.

In this talk I will explain how to extend the notions of Schwartz functions (or rather measures) from smooth, semi-algebraic ("Nash") manifolds, to the corresponding category of stacks. I will also describe an approach to orbital integrals (in this setting considered as "evaluation maps" on global Schwartz spaces) which does not use truncation.