The algebra of invariants of the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra

TL;DR

This paper investigates the algebra of invariants under the adjoint action of the unitriangular group on the nilradical of a parabolic subalgebra, proposing a conjecture and proving it for specific cases.

Contribution

It introduces a conjecture on the structure of the invariant algebra and proves it for certain types of parabolic subalgebras.

Findings

Conjecture on the structure of the invariant algebra

Proof of the conjecture for specific parabolic subalgebras

Advances understanding of invariants in Lie algebra representations

Abstract

In the paper the algebra of invariants of the adjoint action of the unitriangular group in the nilradical of a parabolic subalgebra is studied. We set up a conjecture on the structure of the algebra of invariants. The conjecture is proved for parabolic subalgebras of special types.