Notes on Atkin-Lehner theory for Drinfeld modular forms

In this article, we settle a part of the Conjecture by Bandini and Valentino (\cite{BV19a}) for $S_{k,l}(\Gamma_0(T))$ when $\mathrm{dim}\ S_{k,l}(\mathrm{GL}_2(A))\leq 2$. Then, we frame this conjecture for prime, higher levels, and provide some evidence in favour of it. For any square-free level $\mathfrak{n}$, we define oldforms $S_{k,l}^{\mathrm{old}}(\Gamma_0(\mathfrak{n}))$, newforms $S_{k,l}^{\mathrm{new}}(\Gamma_0(\mathfrak{n}))$, and investigate their properties. These properties depend on the commutativity of the (partial) Atkin-Lehner operators with the $U_\mathfrak{p}$-operators. Finally, we show that the set of all $U_\mathfrak{p}$-operators are simultaneously diagonalizable on $S_{k,l}^{\mathrm{new}}(\Gamma_0(\mathfrak{n}))$.