Submodule Lattice of Generalized Verma Modules

Abstract

Let 𝒢 be a simple finite dimensional Lie algebra over the complex numbers and let 𝒢¯ = 𝒢₁ ⊕…⊕ 𝒢 _k be a regular semisimple subalgebra of 𝒢 with each 𝒢 _i being a simple algebra of type A or C. It is shown that the lattice of submodules of a generalized Verma 𝒢-module constructed by parabolic induction starting from a simple torsion free 𝒢¯-module is almost always isomorphic to the lattice of submodules of an associated module formed as a quotient of a classical Verma module by a sum of Verma submodules. In particular, it is shown that the Mathieu admissible Verma modules involved have maximal submodules which are the sum of Verma modules.

Key Words:

• Lie algebras
• Verma modules
• Generalized Verma modules
• Bounded multiplicities
• Lattice of submodules

Acknowledgments

The authors would like to thank the referee for several helpful comments, in particular for suggesting the elegant proof of Lemma 8 for the A _n case.

D. J. Britten was supported in part by NSERC Grant #OGP0008471. V. M. Futorny was supported in part by CNPQ #300679/1997-1. F. W. Lemire was supported in part by NSERC Grant #OGP0007742.