An analytic separation of series of representations for ${\rm SL}(2;\mathbb R)$

For the group ${\rm SL}(2;\mathbb R)$, holomorphic wave fronts of the projections on different series of representations are contained in some disjoint cones. These cones are convex for holomorphic and antiholomorphic series, which corresponds to the well-known fact that these projections can be extended holomorphically to some Stein tubes in ${\rm SL}(2;\mathbb C)$. For the continuous series, the cone is not convex, and the projections are boundary values of 1-dimensional $\bar\partial$-cohomology in a non-Stein tube.