Index of Twisted Elliptic Boundary Value Problems Associated with Isometric Group Actions

Abstract Given an isometric action of a discrete group $$\Gamma$$ on a compact manifold $$M$$ with boundary and a $$\Gamma$$ -invariant elliptic boundary value problem $$\mathcal{D}$$ on $$M$$ , we consider its twisting by projections over the crossed product algebra $$C^{\infty}(M)\rtimes\Gamma$$ . The twisted problem is Fredholm and we compute its index in terms of the equivariant Chern character of the principal symbol of $$\mathcal{D}$$ and a noncommutative Chern character of $$P$$ . In the special case, when $$\mathcal{D}$$ is the Dirichlet problem for the Euler operator, the index is expressed as a linear combination of the Euler characteristics of the fixed point submanifolds of the group action.