Boundary value problems and partial group actions

Let $\mathrm{\Gamma }$ be a discrete group which acts on a manifold $N$. Suppose $M\subseteq N$ is a submanifold which is not $\mathrm{\Gamma }$-invariant and has a boundary. There are partial shifts $U_g$ for $g\in \mathrm{\Gamma }$ on $L^2(M)$ defined by $$U_g\phi (x)=\{\begin{array}{ll}\phi (g^{-1}\cdot x)& \text{if }{g}^{-1}\cdot x\in M,\\ 0& \text{else.}\end{array}$$
In this talk, I will describe an algebra of operators generated by boundary value problems on $M$ and the partial shifts $U_g$ for $g\in \mathrm{\Gamma }$ (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.