---
abstract: 'Let $M$ be a foliated manifold and $G$ a discrete group acting on $M$ by diffeomorphisms mapping leaves to leaves. Then $G$ naturally acts by automorphisms on the algebra of Heisenberg pseudodifferential operators on the foliation. Our main result is an index theorem for hypoelliptic-type operators which belong to the crossed product of the Heisenberg pseudodifferential operators with the group $G$. As a corollary we get an explicit formula, in terms of characteristic classes of equivariant vector bundles over $M$, for the Chern-Connes character associated to the hypoelliptic signature operator constructed by Connes and Moscovici.'
address: |
    Université de Lyon\
    CNRS UMR 5208\
    Université Lyon 1\
    Institut Camille Jordan\
    43, Bd du 11 novembre 1918, 69622 Villeurbanne Cedex, France
author:
- 'Denis Perrot, Rudy Rodsphon'
title: An equivariant index theorem for hypoelliptic operators
---

<span style="font-variant:small-caps;">Keywords.</span> Cyclic cohomology, K-theory, Index theory, Pseudodifferential operators\
<span style="font-variant:small-caps;">MSC.</span> 19D55, 19K56, 58J42, 46L87

Introduction
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This paper deals with the index theory of certain equivariant hypoelliptic operators on foliated manifolds. Our motivation partly comes from the Connes-Moscovici *transverse* index theorem in [@CM1995]. We let $(M, V)$ be a (possibly non-compact) foliated manifold. To such a foliation is canonically associated the algebra $\Psi_{H,c}(M)$ 