On the Mathieu Conjecture for $Sp(N)$ and G₂

As a direct continuation of K. Zwart, arXiv:2304.02648, which is built on the work of M. M\"uger and L. Tuset, we reduce the Mathieu conjecture, formulated by O. Mathieu in 1997, for $Sp(N)$ and G₂ to a conjecture involving functions over Rⁿ× (S¹)ᵐ with n,m₀. The proofs rely on Euler-style parametrizations of these groups, a specific version of the $KAK$ decomposition, which we discuss and prove.