Extremal Derived Length in the Unipotent Radicals of Cartan-Type Automorphism Groups

Abstract

Over perfect fields of characteristic $p\ge5$, the unipotent radicals of the height-one special, Witt-Jacobson, and Hamiltonian automorphism groups in at least three, two, and four variables, respectively, attain the largest derived length allowed by their nilpotency classes. More strongly, every term of each derived series equals the corresponding power-of-two term of the lower central series. These equalities hold on finite-field points, with explicit Frattini quotients and lower $p$-central series. For the special family, a Cartier quotient measures the difference between the congruence and lower central filtrations. The Hamiltonian correction comes from flux and the socle line; the Witt-Jacobson family has none. We also identify the contact automorphism group scheme and, at arbitrary divided-power height, compute the Cartier-Verschiebung maps on restricted derivations and the Verschiebung filtration and distribution algebra of the Frobenius thickening of the special radical.