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

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.