L 2-Lefschetz distribution of Lie foliations

Abstract

Let $${\fancyscript{F}}$$ be a Lie foliation on a closed manifold M with structural Lie group G. Let $${\rho : \left(\widetilde{M}, \widetilde{\fancyscript{F}}\right) \to (M,\fancyscript{F})}$$ a regular covering map with group of deck transformations Γ. A distribution $${L_{\Gamma {\rm dis}} (\fancyscript{F})}$$ on G is defined. This $${L_{\Gamma {\rm dis}} (\fancyscript{F})}$$ is an L 2 version of the Lefschetz distribution of $${\fancyscript{F}}$$ , introduced by Alvarez-Kordyukov. Around the identity element, a distributional version of the Atiyah’s Γ-index theorem is proved.