An analogue of the Kostant-Rallis multiplicity theorem for $\theta$-group harmonics

TL;DR

This paper generalizes the Kostant-Rallis multiplicity theorem to --groups, providing explicit formulas for key examples involving E6 and E8, advancing understanding in representation theory of symmetric spaces.

Contribution

It extends the Kostant-Rallis multiplicity formula to --groups, including explicit cases for E6 and E8, which were previously not known.

Findings

Explicit formulas for E6 (three qubits) case

Explicit formulas for E8 case

Generalization of multiplicity theorem to --groups

Abstract

The main result in this paper is the generalization of the Kostant-Rallis multiplicity formula to general {\theta}--groups (in the sense of Vinberg). The special cases of the two most interesting examples one for $E_6$ (three qubits) and one for $E_8$ are given explicit formulas.