An index theorem for gauge-invariant families: The case of solvable groups

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, ψ_inv^∞(Y) and Ψ_inv^∞(Y), are generalizations of "parameter dependent" algebras of pseudodifferential operators (with parameter in 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.