Modular forms for $\mathrm{G}\mathrm{L}(r,{\mathbb{F}}_q[T])$: Hecke operators and growth of expansion coefficients

We determine the action of the Hecke operators $T_{\mathfrak{p},i}$ on the coefficient forms $g_1,\dots ,g_{r-1},g_r=\mathrm{\Delta }$, and $h$, which together generate the ring of modular forms for $\mathrm{G}\mathrm{L}(r,{\mathbf{F}}_q[T])$. All these are eigenforms with powers of $\pi$ as eigenvalues, where $\pi$ is the monic generator of the prime ideal $\mathfrak{p}$ of ${\mathbb{F}}_q[T]$. We further describe the growth of the $t$-expansion coefficients of the discriminant function $\mathrm{\Delta }$. It is such that the product expansion of $\mathrm{\Delta }$ as well as the $t$-expansion of each modular form converges on the natural fundamental domain for $\mathrm{G}\mathrm{L}(r,{\mathbf{F}}_q[T])$.