Cartan subalgebras for restrictions of $\mathfrak{g}$-modules
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.
- Publication:
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arXiv e-prints
- Pub Date:
- May 2024
- DOI:
- 10.48550/arXiv.2405.10382
- arXiv:
- arXiv:2405.10382
- Bibcode:
- 2024arXiv240510382K
- Keywords:
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- Mathematics - Representation Theory
- E-Print:
- 32 pages, modify Definition 3.29 and add Subsection 3.5