Cartan subalgebras for restrictions of $\mathfrak{g}$-modules

TL;DR

This paper introduces a new way to define Cartan subalgebras for certain $rak{g}$-modules, linking algebraic support structures with geometric and representation-theoretic concepts.

Contribution

It proposes a novel definition of Cartan subalgebras for $rak{g}$-modules based on support analysis, connecting algebraic and geometric perspectives.

Findings

Support of the $rak{Z}(rak{g})$-action forms unions of affine subspaces

Supports are algebraic counterparts of measure supports in unitary decompositions

Defines Cartan subalgebras for modules using support structures

Abstract

In this paper, we deal with the $\mathcal{U}(\mathfrak{g})$-action on a $\mathfrak{g}$-module on which a larger algebra $\mathcal{A}$ acts irreducibly. Under a mild condition, we will show that the support of the $\mathcal{Z}(\mathfrak{g})$-action is a union of affine subspaces in the dual of a Cartan subalgebra modulo the Weyl group action. As a consequence, we propose a definition of a Cartan subalgebra for such a $\mathfrak{g}$-module. The support of the $\mathcal{Z}(\mathfrak{g})$-module is an algebraic counterpart of the support of the measure in the irreducible decomposition of a unitary representation. This consideration is motivated by the theory of the discrete decomposability initiated by T. Kobayashi. Defining a Cartan subalgebra for a $\mathfrak{g}$-module is motivated by the study of I. Losev on Poisson $G$-varieties. These are related each other through the associated variety and the nilpotent orbit associated to a $\mathfrak{g}$-module.