The Mathieu Property for Compact Connected Lie Groups

Abstract

Let $G$ be a compact connected Lie group, let $\mathcal{R}(G)$ denote its algebra of representative functions, and let ${\mathcal{I}}_G(f)={\int }_{G}f(g)\mathrm{d}g$ be normalized Haar integration. We prove that $\ker {\mathcal{I}}_G$ is a Mathieu--Zhao subspace of $\mathcal{R}(G)$ if and only if $G$ is a torus. More strongly, for every nonabelian compact connected Lie group $G$ we construct $A,P,Q\in \mathcal{R}(G)$, with $A\ge 0$ and $A\not\equiv 0$, such that all pure moments of $P$ vanish and the marked moments satisfy an exact Pascal-row identity $${\mathcal{I}}_G(Q^s{P}^m)=c_m\left(\genfrac{}{}{0ex}{}{m-1}{s-1}\right){\mathcal{I}}_G(A^{4m+s})>0(1\le s\le m),$$ where $c_m=4^m(m!{)}^2/(2m+1)!$; the same moments vanish for $s>m$. The construction begins with a homogeneous, phase-balanced polynomial pair on ${\mathbb{C}}^2$. An exact coefficient identity on the Hopf sphere $S^3$ is transferred by orbit averaging to every compactly supported $SU(2)$-invariant measure on ${\mathbb{C}}^2$. A highest-weight representation and a visible simple-root doublet produce such a measure on every compact simple Lie group. Phase balance gives descent through every central quotient, and pullback from an adjoint simple quotient handles arbitrary nonabelian compact connected groups. The proof is direct and independent of the implication from the Mathieu conjecture to the Jacobian conjecture. The torus direction is the theorem of Duistermaat and van der Kallen.

arXiv comment14 pages, 1 table; Python program is on GitHub