Modular forms for \(\mathrm{GL}(r, \mathbb{F}_{q}[T])\): Hecke operators and growth of expansion coefficients

We determine the action of the Hecke operators $T_{\mathfrak{p},i}$ on the coefficient forms $g_{1}, \dots, g_{r-1}, g_{r} = \Delta$, and $h$, which together generate the ring of modular forms for $\mathrm{GL}(r, \mathbf{F}_{q}[T])$. 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 $\Delta$. It is such that the product expansion of $\Delta$ as well as the $t$-expansion of each modular form converges on the natural fundamental domain for $\mathrm{GL}(r, \mathbf{F}_{q}[T])$.