On the Mathieu Conjecture for $Sp(N)$ and $G_2$

As a direct continuation of K. Zwart, arXiv:2304.02648, which is built on the work of M. Müger and L. Tuset, we reduce the Mathieu conjecture, formulated by O. Mathieu in 1997, for $Sp(N)$ and $G_2$ to a conjecture involving functions over ${\mathbb{R}}^n\times (S^1{)}^m$ with $n,m\in {\mathbb{N}}_0$. The proofs rely on Euler-style parametrizations of these groups, a specific version of the $KAK$ decomposition, which we discuss and prove.