Metabelian Lie powers of the natural module for a general linear group

Consider a free metabelian Lie algebra M of finite rank r over an infinite field K of prime characteristic p. Given a free generating set, M acquires a grading; its group of graded automorphisms is the general linear group GL _r(K), so each homogeneous component M ^d is a finite dimensional GL _r(K)-module. The homogeneous component M ¹ of degree 1 is the natural module, and the other M ^d are the metabelian Lie powers of the title.This paper investigates the submodule structure of the M ^d. In the main result, a composition series is constructed in each M ^d and the isomorphism types of the composition factors are identified both in terms of highest weights and in terms of Steinberg's twisted tensor product theorem; their dimensions are also given. It turns out that the composition factors are pairwise non-isomorphic, from which it follows that the submodule lattice is finite and distributive. By the Birkhoff representation theorem, any such lattice is explicitly recognizable from the poset of its join-irreducible elements. The poset relevant for M ^d is then determined by exploiting a 1975 paper of Yu.A. Bakhturin on identical relations in metabelian Lie algebras.