The multiplicity of the Steinberg representation of ${\rm GL}\sb n{\bf F}\sb q$ in the symmetric algebra

Let $ S(V)$-dimensional representation $ V$ $ {\text{G}}{{\text{L}}_n}{{\mathbf{F}}_q}$The multiplicity series in $ S(V)$ $ {\text{G}}{{\text{L}}_n}{{\mathbf{F}}_q}$ $ {F_{{\text{S}}\text{t}}}(t) = \sum\nolimits_{k = 0}^\infty {{a_k}{t^k}} $ is the multiplicity of St in the $ k$. We show that $ {F_{{\text{S}}t}}(t) = {t^r}\prod\nolimits_{i = 1}^n {{{(1 - {t^{{q^i} - 1}})}^{ - 1}}} $ $ r = \sum\nolimits_{i = 1}^{n - 1} {({q^i} - 1} )$ of the parabolic subgroups of <IMG WIDTH="68" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img18.gif" ALT="$ {\text{G}}{{\text{L}}_n}{{\mathbf{F}}_q}$">.