Matrix valued commuting differential operators with $A_2$ symmetry

TL;DR

This paper explicitly constructs matrix-valued commuting differential operators with $A_2$ symmetry on a homogeneous vector bundle over a symmetric space, generalizing previous operators with respect to parameters and potential functions.

Contribution

It provides explicit computations of radial parts of generators and introduces a generalization of commuting differential operators with new parameters and potentials.

Findings

Explicit matrix-valued commuting differential operators with $A_2$ symmetry

Radial parts of generators computed explicitly

Generalization with respect to parameters and potential functions

Abstract

We study the algebra of invariant differential operators on a certain homogeneous vector bundle over a Riemannian symmetric space of type $A_2$. We computed radial parts of its generators explicitly to obtain matrix-valued commuting differential operators with $A_2$ symmetry. Moreover, we generalize the commuting differential operators with respect to a parameter and the potential function.