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 $\wp\in A$ be a monic irreducible polynomial of positive degree. Let $k\geq 2$ and $r\geq 1$ be integers. Let $S_k(\Gamma_1(\mathfrak{n}\wp^r))$ be the space of Drinfeld cuspforms of level $\Gamma_1(\mathfrak{n}\wp^r)$ and weight $k$. In this paper, we prove that the multiplicity of a Hecke eigensystem of finite $\wp$-slope in $S_k(\Gamma_1(\mathfrak{n}\wp^r))$ is equal to $q^{(r-1)\mathrm{deg}(\wp)}$ times that in $S_k(\Gamma_1(\mathfrak{n}\wp))$. In particular, this shows that a Hecke eigensystem of finite $\wp$-slope appears in $S_k(\Gamma_1(\mathfrak{n}\wp^r))$ if and only if it appears in $S_k(\Gamma_1(\mathfrak{n}\wp))$.