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

Abstract

We define the gauge-equivariant index of a family of elliptic operators invariant with respect to the free action of a family \(\mathcal{G} \to B\) of Lie groups (these families are called ``gauge-invariant families'' in what follows). If the fibers of \(\mathcal{G} \to B\) 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 on the bundle \(\mathcal{G} \to B\). The algebras of invariant pseudodifferential operators that we study, \(\psi _{{\text{inv}}}^\infty (Y)\) and \({\psi }_{{inv}}^\infty (Y)\), 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.