The branching problem for generalized Verma modules, with application to the pair $(so(7),Lie G_2)$, extended version with tables

TL;DR

This paper investigates the branching problem for generalized Verma modules in reductive Lie algebra pairs, providing explicit character-based formulas and singular vectors, with detailed application to the pair (so(7), G_2).

Contribution

It introduces a method to analyze branching using character projections and center actions, specifically applied to the (G_2, so(7)) pair, including tables and detailed cases.

Findings

Explicit formulas for branching in the (G_2, so(7)) pair.

Construction of singular vectors realizing branching.

Application to various parabolic subalgebras and representations.

Abstract

We discuss the branching problem for generalized Verma modules $M_\lambda(\mathfrak g, \mathfrak p)$ applied to couples of reductive Lie algebras $\bar{\mathfrak g}\hookrightarrow \mathfrak g$. The analysis is based on projecting character formulas to quantify the branching, and on the action of the center of $U(\bar{\mathfrak g})$ to explicitly construct singular vectors realizing part of the branching. We demonstrate the results on the pair $\mathrm{Lie}G_2\hookrightarrow{so(7)}$ for both strongly and weakly compatible with $i(\mathrm {Lie} G_2)$ parabolic subalgebras and a large class of inducing representations.