Measures on differentiable stacks

Abstract

We introduce and study measures and densities (geometric measures) on differentiable stacks, using a rather straightforward generalization of Haefliger’s approach to leaf spaces and to transverse measures for foliations. In general we prove Morita invariance, a Stokes formula which provides reinterpretations in terms of (Ruelle–Sullivan type) algebroid currents, and a Van Est isomorphism. In the proper case we reduce the theory to classical (Radon) measures on the underlying space, we provide explicit (Weyl-type) formulas that shed light on Weinstein’s notion of volumes of differentiable stacks; in particular, in the symplectic case, we prove the conjecture left open in Weinstein’s “The volume of a differentiable stack” \[33].We also revisit the notion of Haar systems (and the existence of cut-off functions). Our original motivation comes from the study of Poisson manifolds of compact types \[8–10], which provide two important examples of such measures: the affine and the Duistermaat–Heckman measures.