Invariant Differential Operators for the Real Exceptional Lie Algebra $F'_4$
V. K. Dobrev
TL;DR
The paper addresses the problem of classifying invariant differential operators for the split real exceptional Lie algebra $F'_4$ by employing parabolic induction from a maximal parabolic. It develops a systematic, representation-theoretic framework using generalized Verma modules induced from $M\oplus A\oplus N$ with $M=sl(3,R)\oplus sl(2,R)$, organizing reducible modules into multiplets and identifying singular vectors that generate the corresponding differential operators; the multiplets are parameterized by four positive integers $[m_1,m_2,m_3,m_4]$ and each contains 96 generalized Verma modules, connected by embeddings with levels $n$. A Knapp–Stein symmetry acts on the conformal weight $d = 7/2 + c$, linking spaces via intertwiners and yielding a structured catalog of invariant differential operators for $F'_4$. The work also discusses the potential appearance of discrete series representations within the induced family, and provides a framework with physical motivation through the chosen parabolic subalgebra and its compact/cuspidal components. Overall, the paper delivers a complete multiplet-based classification of invariant differential operators for the real split form of $F_4$, grounded in explicit generalized Verma module embeddings and their singular vectors.
Abstract
In the present paper we continue the project of systematic construction of invariant differential operators on the example of the non-compact exceptional Lie algebra $F'_4=F_{4(4)}$ which is split real form of the exceptional Lie algebra $F_4$. We consider induction from a maximal parabolic algebra. We classify the reducible Verma modules over $F_4$ which are compatible with this induction. Thus, we obtain the classification of the corresponding invariant differential operators.