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

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