*Organization of the paper.* Section \[sindex\] is a self-contained introduction to the equivariant Heisenberg pseudodifferential calculus on foliated manifolds, the pseudodifferential extension, and the Connes-Moscovici residue. Then Theorem \[intro thm equivariant Radul cocycle\] is proved.\
In sections \[sbimod\], \[scano\] and \[sdirac\] we adapt the formalism of [@Per2012] to the Heisenberg calculus. We introduce various spaces of operators acting on formal Heisenberg symbols, and the universal Dirac operators which will be used in our algebraic JLO formula.\
Section \[sequiv\] introduces the required objects to carry this formalism to the equivariant setting. In particular, we recall the point of view we need to construct Connes’ characteristic map from the equivariant cohomology $H^{\bullet}(EG \times_G M)$ to the periodic cyclic cohomology of the crossed product ${\mathrm{HP}}^{\bullet}({C^{\infty}}(M) \rtimes G)$. From the technical side we use the $X$-complex of Cuntz and Quillen [@CQ95].\
Section \[sJLO\] finally gives the algebraic JLO formula on the algebra of formal (equivariant) Heisenberg symbols, leading to Theorem \[intro thm equivariant index\]. This is again an adaptation of the formalism developed in [@Per2014] for the non-Heisenberg case.\
Section \[CM signature\] shows how to deduce Theorem \[intro thm CM signature\] from Theorem \[intro thm equivariant index\].

Equivariant local index formula {#sindex}
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Heisenberg 