Gelfand-Tsetlin modules for Lie algebras of rank $2$

TL;DR

This work extends the centralizer approach to construct and classify simple weight modules with infinite weight spaces for rank-2 simple Lie algebras by introducing and exploiting $\Gamma$-pointed modules. It provides explicit presentations of Cartan centralizers $U_0(\mathfrak{g})$ for $\mathfrak{sl}_3$, $\mathfrak{so}_5$ (type $C_2$), and $G_2$, and builds families of torsion-free, $\Gamma$-pointed Gelfand-Tsetlin-type modules $V(a_1,a_2,a_3,\dots)$ that are simple under concrete nonvanishing conditions. The results show that these families, and their subquotients, exhaust all generic simple GT-type modules for the rank-2 types studied, and they reveal a coherent decomposition of $G_2$-modules via embedded $A_2$-subalgebras, linking GT theory across types. Overall, the paper broadens GT-module theory beyond type $A$ and provides explicit, computable models suitable for further constructions by parabolic induction to higher-rank or affine algebras.

Abstract

We explicitly construct families of simple modules for Lie algebras of rank $2$, on which certain commutative subalgebra acts diagonally and has a simple spectrum. In type $A$ these modules are well known generic Gelfand-Tsetlin modules and they can be viewed as such for other rank $2$ Lie algebras.

Theorems & Definitions (32)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Theorem 2.5
• proof
• Proposition 4.1
• Theorem 4.2