On Decomposition of Drinfeld cusp forms of level $t$

In this article, we first prove that the Hecke operator $T_t$ has no eigenform in $\text{Sla}$ with eigenvalue $-t^{k\mathrm{/}2}$, when the characteristic of the base field is odd. Furthermore, if $\dim \text{Slt}$ is even, we show that $T_t$ has no eigenform in $\text{Sla}$ with eigenvalue $t^{k\mathrm{/}2}$. As a consequence, we prove that the direct sum decomposition $\text{Slt}=\text{Sold}\oplus \text{Snew}$ holds when $\dim \text{Slt}$ is even. This proves the conjecture \cite[Conjecture 1.1(3)]{BV19a} of Bandini and Valentino for an infinite family of cusp forms. In particular, for any weight $k$, there exists at least one type $m$ (there are only two possible non-trivial values of $m$) for which the conjecture \cite[Conjecture 1.1(3)]{BV19a} is true.