We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family of Lie groups (these families are called ``gauge-invariant families'' in what follows). If the fibers of 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 . The algebras of invariant pseudodifferential operators that we study, and , 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.

group action - non-commutative geometry - elliptic operator - gauge group - index formula - K-theory - family index