Derived right adjoints of parabolic induction: an example

Abstract: Suppose $p \geq 5$ is a prime number, and let $G = \textrm{SL}_2(\mathbb{Q}_p)$. We calculate the derived functors $\textrm{R}^n\mathcal{R}_B^G(\pi)$, where $B$ is a Borel subgroup of $G$, $\mathcal{R}_B^G$ is the right adjoint of smooth parabolic induction constructed by Vign\'eras, and $\pi$ is any smooth, absolutely irreducible, mod $p$ representation of $G$.