Local nilpotence criterion for arbitrary derivations of the first Weyl algebra

Determine whether, for every nonzero derivation of the first Weyl algebra A_1, the existence of automorphisms of arbitrarily large degree in its isotropy group implies that the derivation is locally nilpotent.

Background

The paper studies whether Pan’s characterization of locally nilpotent derivations by the presence of arbitrarily high-degree automorphisms in the isotropy group extends from the polynomial algebra to noncommutative algebras. For the first Weyl algebra A_1, the authors prove the equivalence only for nonzero locally finite derivations: within that class, a derivation is locally nilpotent exactly when its isotropy group contains automorphisms of unbounded degree.

The unresolved issue is whether the same converse holds without the locally finite hypothesis. Equivalently, for an element w in A_1, the paper asks whether an isotropy set consisting of automorphisms that transform w by a scalar translation and have unbounded degree necessarily forces the inner derivation ad_w to be locally nilpotent. The authors state this as a separate formal problem after proving the locally finite case.