Boundary value problems and partial group actions

Let $\Gamma$ be a discrete group which acts on a manifold $N$. Suppose $M\subseteq N$ is a submanifold which is not $\Gamma$-invariant and has a boundary. There are partial shifts $U_g$ for $g\in\Gamma$ on $L^2(M)$ defined by $$ U_g\varphi(x)=\begin{cases} \varphi(g^{-1}\cdot x) &\text{if }g^{-1}\cdot x\in M,\\ 0 &\text{else.} \end{cases}$$
In this talk, I will describe an algebra of operators generated by boundary value problems on $M$ and the partial shifts $U_g$ for $g\in\Gamma$ (under suitable assumptions on the action). As in the classical Boutet de Monvel calculus there are two principal symbol maps: one associated with the interior and one with the boundary. Here, they take values in crossed product algebras of corresponding partial group actions. I will discuss how one can classify the stable homotopy classes of elliptic operators over the considered algebra in terms of $K$-theory.

The talk is based on joint work with Anton Savin and Elmar Schrohe.