The F-method and a branching problem for generalized Verma modules associated to $({\LieGtwo},{so(7)})$

TL;DR

This paper applies the F-method to analyze the branching problem for generalized Verma modules associated with the non-compatible Lie algebras (G2, so(7)), classifying singular vectors in this context.

Contribution

It introduces a novel application of the F-method to the (G2, so(7)) pair, providing a classification of singular vectors for generalized conformal so(7)-Verma modules.

Findings

Classified G2 ∩ p'-singular vectors for so(7)-modules.

Applied the F-method to non-compatible Lie algebra pairs.

Extended understanding of branching problems in representation theory.

Abstract

The branching problem for a couple of non-compatible Lie algebras and their parabolic subalgebras applied to generalized Verma modules was recently discussed in \cite{ms}. In the present article, we employ the recently developed F-method, \cite{KOSS1}, \cite{KOSS2} to the couple of non-compatible Lie algebras $({\LieGtwo},{so(7)})$, and generalized conformal ${so(7)}$-Verma modules of scalar type. As a result, we classify the $\LieGtwo \cap \gop'$-singular vectors for this class of $so(7)$-modules.