On globally hypoelliptic abelian actions and their existence on homogeneous spaces

The candidate solution reproduces the paper’s two main results (Theorem 1 and Theorem 2) and follows the same proof strategy. For part (a), it invokes the GH ⇒ only smooth invariant distributions property (Proposition 2.1) and then constructs non-smooth invariant distributions by embedding the R^2-action into subgroups involving SL(2,R), exactly as in the paper’s two-case analysis (two unipotent generators, or one partially hyperbolic and one quasi-unipotent) leading to Proposition 3.2 and Theorem 4; hence GH is impossible on SL(n,R)/D with a quasi-unipotent generator . For part (b), it uses the same reduction: finite invariant volume implies compactness for solvable quotients, an ergodic quasi-unipotent generator forces class (I), and any ergodic homogeneous flow on a compact class-(I) solvmanifold is smoothly conjugate to a homogeneous flow on a compact nilmanifold; the conjugacy carries the entire commuting R^k-action (as in the paper’s proof of Theorem 2) . Minor clarifications (e.g., the quasi-unipotent⇒ad-nilpotent step in sl(n,R), and explicit handling of the joint-invariance construction) would strengthen the candidate write-up, but there is no substantive discrepancy.