On some constancy of Hecke eigensystems for Drinfeld cuspforms of finite slope

Let $p$ be a rational prime, let $q>1$ be a $p$-power integer, let ${\mathbb{F}}_q$ be the field of $q$ elements and let $A={\mathbb{F}}_q[t]$ be the polynomial ring over ${\mathbb{F}}_q$. Let $\mathfrak{n}\in A$ be a nonzero element and let $\mathrm{\wp }\in A$ be a monic irreducible polynomial of positive degree. Let $k\ge 2$ and $r\ge 1$ be integers. Let $S_k({\mathrm{\Gamma }}_1(\mathfrak{n}{\mathrm{\wp }}^r))$ be the space of Drinfeld cuspforms of level ${\mathrm{\Gamma }}_1(\mathfrak{n}{\mathrm{\wp }}^r)$ and weight $k$. In this paper, we prove that the multiplicity of a Hecke eigensystem of finite $\mathrm{\wp }$-slope in $S_k({\mathrm{\Gamma }}_1(\mathfrak{n}{\mathrm{\wp }}^r))$ is equal to $q^{(r-1)\mathrm{d}\mathrm{e}\mathrm{g}(\mathrm{\wp })}$ times that in $S_k({\mathrm{\Gamma }}_1(\mathfrak{n}\mathrm{\wp }))$. In particular, this shows that a Hecke eigensystem of finite $\mathrm{\wp }$-slope appears in $S_k({\mathrm{\Gamma }}_1(\mathfrak{n}{\mathrm{\wp }}^r))$ if and only if it appears in $S_k({\mathrm{\Gamma }}_1(\mathfrak{n}\mathrm{\wp }))$.