  We define the gauge-equivariant index of a family of elliptic operators
invariant with respect to the free action of a family $\GR \to B$ of Lie groups
(these families are called ``gauge-invariant families'' in what follows). If
the fibers of $\GR \to B$ are simply-connected and solvable, we compute the
Chern character of the gauge-equivariant index, the result being given by an
Atiyah-Singer type formula that incorporates also topological information about
the bundle $\GR \to B$. The algebras of invariant pseudodifferential operators
that we study, $\Psm {\infty}Y$ and $\PsS {\infty}Y$, are generalizations of
``parameter dependent'' algebras of pseudodifferential operators (with
parameter in $\mathbb R^q$), so our results provide also an index theorem for
elliptic, parameter dependent pseudodifferential operators. We apply these
results to study Fredholm boundary conditions on a simplex.
