On the $\mathrm{\ell }$-modular composition factors of the Steinberg representation

Let $G$ be a finite group of Lie type and ${\text{St}}_k$ be the Steinberg representation of $G$, defined over a field $k$. We are interested in the case where $k$ has prime characteristic~$\mathrm{\ell }$ and ${\text{St}}_k$ is reducible. Tinberg has shown that the socle of ${\text{St}}_k$ is always simple. We give a new proof of this result in terms of the Hecke algebra of $G$ with respect to a Borel subgroup and show how to identify the simple socle of ${\text{St}}_k$ among the principal series representations of~$G$. Furthermore, we determine the composition length of ${\text{St}}_k$ when $G={\text{GL}}_n(q)$ or $G$ is a finite classical group and $\mathrm{\ell }$ is a so-called linear prime.