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 [33]. We also revisit the notion of modular class and of Haar systems (and the existence of cut-off functions). Our original motivation comes from the study of Poisson manifolds of compact types [8,9,10], which provide two important examples of such measures: the affine and the Duistermaat-Heckman measures.

The modular class of a singular foliation

“…Modular classes of Lie algebroids have been defined as the natural counterpart of the notion of modular vector fields of Poisson manifolds [29]. It is conjectured that it measures the obstruction of the existence of a volume form on the differentiable stack associated to the groupoid integrating the Lie algebroid [7,30]. The generalization of the notion of modular class to the Lie ∞-algebroid context has already been investigated in [5] and we merely summarize some of their result here, with notations adapted to the present paper.…”

“…The modular class of Lie algebroids also restricts to the well-known notion of modular class of regular foliations when the Lie algebroid in question is a foliation Lie algebroid. This allows to think about the modular class of a Lie algebroid as an obstruction to the existence of a certain invariant measure on the differentiable stack associated to this Lie algebroid [7,30].…”

Sheaves, principal bundles, and Čech cohomology for diffeological spaces

“…By [5,Proposition 10.6], i * G admits a proper normalized Haar density ρ; that is, a smooth family of normalized densities on the fibers of s for which s| t −1 (supp(ρ)) : t −1 (supp(ρ)) → N (U) 0 is proper and which is invariant under right multiplication by elements of (i * G) 1 (see [5,Section 10] for details):…”

On evolution equations for Lie groupoids

“…where D tr AG is the transverse density bundle of AG [7]. The convolution algebra Throughout this paper we make the convention:…”

“…as well as straight properties of the calculus of densities. To finish with this description, we mention that Ω 1/2 0 is related, but distinct, to the transverse density bundle D tr AG of [7]. The latter is G-invariant and serves to produce geometric transverse measures useful for the geometry of groupoids and stacks, while our choice of "transverse" bundle is required for the pairing with densities in Ω 1/2 , but only equivariant with respect to R-actions provided by invariant vectors fields.…”