An addendum on the Mathieu Conjecture for $SU(N)$, $Sp(N)$ and $G_2$
Kevin Zwart
TL;DR
The paper sharpens the Mathieu Conjecture for $SU(N)$ by employing a generalized Euler angles/KAK decomposition to reduce nonabelian integrals to abelian torus-type data, and it reformulates the problem in terms of Laurent polynomials on $\mathbb{R}^n\times (S^1)^m$. It provides an explicit inductive parameterization $F_{SU(N)}$, an associated Haar measure, and a recursive construction of finite-type functions $f^{SU(N)}$, yielding integral factorizations with Jacobians $\tilde{J}_{SU(N)}$. Under Conjecture $\mathrm{con:xz}$ (and its variants for $Sp(N)$ and $G_2$), it establishes a convex-hull criterion on the spectrum ensuring the Mathieu Conjecture for these groups, and shows that no $N$-th roots are needed, allowing a $\,\mathbb{T}^n$-based analysis. The approach further extends to $Sp(N)$ and $G_2$ by analogous decompositions and Jacobians, thereby broadening the scope of abelian-reduction techniques in this domain. Overall, the work strengthens the bridge between nonabelian harmonic analysis on compact Lie groups and abelian, torus-based methods for the Mathieu Conjecture, under standard conjectural assumptions.
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.