Prolongation rigidity of sub-free Lie algebras

Abstract

We prove that if the 0-th Tanaka prolongation $\mathfrak{g}_0=\mathfrak{der}_0(\mathfrak{m})$ of a fundamental graded nilpotent Lie algebra $\mathfrak{m}=\mathfrak{g}_{-s}\oplus\dots\oplus\mathfrak{g}_{-1}$ is irreducible on $\mathfrak{g}_{-1}$, then $\mathfrak{m}$ is prolongation rigid: $\text{pr}_+(\mathfrak{m})=0$. The only exceptions are given by negative gradations of maximal parabolic subalgebras of a simple Lie algebra.