An addendum on the Mathieu Conjecture for $SU(N)$, $Sp(N)$ and $G_2$

Abstract

In this paper, we sharpen results obtained by the author in 2023. The new results reduce the Mathieu Conjecture on $SU(N)$ (formulated for all compact connected Lie groups by O. Mathieu in 1997) to a conjecture involving only functions on ${\mathbb{R}}^n\times (S^1{)}^m$ with $n,m$ non-negative integers instead of involving functions on ${\mathbb{R}}^n\times (S^1\setminus \{1\}{)}^m$. The proofs rely on a more recent work of the author (2024) and a specific $KAK$ decomposition. Finally, with these results we can also improve the results on the groups $Sp(N)$ and $G_2$ in the latter paper, since they relied on the construction introduced in the 2023 paper.

13 pages, no figures