Cartan subalgebras for restrictions of g-modules

In this paper, we deal with the U(g)-action on a g-module on which a larger algebra A acts irreducibly. Under a mild condition, we will show that the support of the Z(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 g-module. The support of the Z(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 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 g-module.