Cygan–Richardson criterion for globally hypoelliptic systems on nilmanifolds

Let $N$ be a nilpotent Lie group of step $r$ with Lie algebra $\mathfrak N$, let $\mathfrak N_j=[\mathfrak N,\mathfrak N_{j-1}]$ for $j=1,\dots,r$, let $\Gamma\setminus N$ be a compact nilmanifold, and let $\{X_1,\dots,X_k\}$ be constant-coefficient vector fields generating an action. The system is globally hypoelliptic (GH) if every distributional solution of $X_1u=f_1,\dots,X_ku=f_k$ with smooth $f_i$ is smooth. Let $\mathfrak L$ be the Lie subalgebra spanned by the $X_i$, and call $\lambda\in\mathfrak N_j^*$ integral when $\lambda(\log\Gamma\cap\mathfrak N_j)\subset\mathbb Z$. Cygan–Richardson conjecture. The system $\{X_1,\dots,X_k\}$ is GH if and only if: (i) it is GH on the associated torus; and (ii), for every nonzero integral functional $\lambda\in(\mathfrak N_j/\mathfrak N_{j+1})^*$,

$$\lambda(\mathfrak L\cap\mathfrak N_j+\mathfrak N_{j+1})\ne0,\qquad j=1,\dots,r-1.$$

The conjecture is proved under additional representation-theoretic hypotheses, and in particular for 2-step nilmanifolds and higher-step nilmanifolds whose coadjoint orbits are all flat; without those hypotheses, the source presents it as unresolved.