Heisenberg Parabolic Subgroups of Exceptional Noncompact $G_{2(2)}$ and Invariant Differential Operators

Abstract: In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact algebra $G_{2(2)}$. We use both the minimal and the maximal Heisenberg parabolic subalgebras. We give the main multiplets of indecomposable elementary representations. This includes the explicit parametrization of the invariant differential operators between the ERs. These are new results applicable in all cases when one would like to use $G_{2(2)}$ invariant differential operators.