Functorial properties of generalised Steinberg representations

Let $G$ be the $F$-points of a connected reductive group over a non-archimedean local field $F$ of residue characteristic $p$ and $R$ be a commutative ring. Let $P=LU$ be a parabolic subgroup of $G$ and $Q$ be a parabolic subgroup of $G$ containing $P$. We study the functor ${\mathrm{S}\mathrm{t}}_Q^G$ taking a smooth $R$-representation $\sigma$ of $L$ which extends to a representation ${\mathrm{e}}_G(\sigma )$ of $G$ trivial on $U$ to the smooth $R$-representation ${\mathrm{e}}_G(\sigma ){\otimes }_R{\mathrm{S}\mathrm{t}}_Q^G(R)$ of $G$ where ${\mathrm{S}\mathrm{t}}_Q^G(R)$ is the generalised Steinberg representation.