Solutions of D_\alpha - 0 from Homogeneous Invariant Functions

TL;DR

This paper establishes a link between homogeneous invariants of semi-simple Lie group representations and solutions to a specific differential equation, showing that invariants guarantee non-trivial solutions.

Contribution

It proves that the existence of a homogeneous invariant implies the existence of non-trivial solutions to D_{α} = 0, connecting invariant theory with differential equations.

Findings

Homogeneous invariants guarantee solutions of D_{α} = 0.

Solutions correspond to maximum invariant ratios.

The work bridges invariant functions and differential equations in Lie theory.

Abstract

We prove that the existence of a homogeneous invariant of degree n for a representation of a semi-simple Lie group guarantees the existence of non-trivial solutions of D_{\alpha} = 0: these correspond to the maximum value of the square of the invariant divided by the norm of the representation to the n^{th} power.