We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family script G sign → B of Lie groups (these families are called "gauge-invariant families" in what follows). If the fibers of G → 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 on the bundle script G sign → B. The algebras of invariant pseudodifferential operators that we study, ψinv∞(Y) and Ψinv∞(Y), are generalizations of "parameter dependent" algebras of pseudodifferential operators (with parameter in Rq). 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.
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