The multiplicity of the Steinberg representation of $\textrm {GL}_ n\textbf {F}_ q$ in the symmetric algebra

Abstract:

Let $S(V)$ denote the symmetric algebra on the standard $n$-dimensional representation $V$ of ${\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}$. The multiplicity series in $S(V)$ for the Steinberg representation St of ${\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}$ is determined. This series is defined by ${F_{{\text {S}}\text {t}}}(t) = \sum \nolimits _{k = 0}^\infty {{a_k}{t^k}}$ where $a_k$ is the multiplicity of St in the $k$th symmetric power ${S^k}(V)$. We show that ${F_{{\text {S}}t}}(t) = {t^r}\prod \nolimits _{i = 1}^n {{{(1 - {t^{{q^i} - 1}})}^{ - 1}}}$, where $r = \sum \nolimits _{i = 1}^{n - 1} {({q^i} - 1} )$. The proof involves a general property of Tits buildings and a computation of the invariants in $S(V)$ of the parabolic subgroups of ${\text {G}}{{\text {L}}_n}{{\mathbf {F}}_q}$.