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

Let $G$ be a finite group of Lie type and $\St_k$ be the Steinberg representation of $G$, defined over a field $k$. We are interested in the case where $k$ has prime characteristic~$\ell$ and $\St_k$ is reducible. Tinberg has shown that the socle of $\St_k$ 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 $\St_k$ among the principal series representations of~$G$. Furthermore, we determine the composition length of $\St_k$ when $G=\GL_n(q)$ or $G$ is a finite classical group and $\ell$ is a so-called linear prime.