  We study from an algebraic point of view the question of extending an action
of a group \(\Gamma\) on a commutative domain \(R\) to a formal
pseudodifferential operator ring \(B=R(\!(x\,;\,d)\!)\) with coefficients in
\(R\), as well as to some canonical quadratic extension
\(C=R(\!(x^{1/2}\,;\,\frac 12 d)\!)_2\) of \(B\). We give a necessary and
sufficient condition of compatibility between the action and the derivation $d$
of $R$ for such an extension to exist, and we determine all possible extensions
of the action to \(B\) and \(C\). We describe under suitable assumptions the
invariant subalgebras \(B^\Gamma\) and \(C^\Gamma\) as Laurent series rings
with coefficients in \(R^\Gamma\).
  The main results of this general study are applied in a numbertheoretical
context to the case where \(\Gamma\) is a subgroup of \({\rm SL}(2,\C)\) acting
by homographies on an algebra \(R\) of functions in one complex variable.
Denoting by \(M_j\) the vector space of algebraic modular forms in $R$ of
weight \(j\) (even or odd), we build for any nonnegative integer \(k\) a linear
isomorphism between the subspace \(C_k^\Gamma\) of invariant operators of order
\(\geq k\) in \(C^\Gamma\) and the product space \(\mathcal{M}_k=\prod_{j\geq
k}M_j\), which can be identified with a space of algebraic Jacobi forms of
weight \(k\). It results in particular a structure of noncommutative algebra on
\(\mathcal M_0\) and an algebra isomorphism \(\Psi:\mathcal M_0\to
C_0^\Gamma\), whose restriction to the particular case of even weights was
previously known in the litterature. We study properties of this correspondence
combining arithmetical arguments and the use of the algebraic results of the
first part of the article.
