On the Cartan-Helgason theorem for supersymmetric pairs
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916706806398976 |
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| author | Sherman, Alexander |
| author_facet | Sherman, Alexander |
| contents | Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac-Moody superalgebra $\mathfrak{g}$. An important branching problem is to determine the finite-dimensional highest-weight $\mathfrak{g}$-modules which admit a $\mathfrak{k}$-coinvariant, and thus appear as functions in a corresponding supersymmetric space $\mathcal{G}/\mathcal{K}$. This is the super-analogue of the Cartan-Helgason theorem. We solve this problem via a rank one reduction and an understanding of reflections in singular roots, which generalize odd reflections in the theory of Kac-Moody superalgebras. An explicit presentation of spherical weights is provided for every pair when $\mathfrak{g}$ is indecomposable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19145 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Cartan-Helgason theorem for supersymmetric pairs Sherman, Alexander Representation Theory Mathematical Physics Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac-Moody superalgebra $\mathfrak{g}$. An important branching problem is to determine the finite-dimensional highest-weight $\mathfrak{g}$-modules which admit a $\mathfrak{k}$-coinvariant, and thus appear as functions in a corresponding supersymmetric space $\mathcal{G}/\mathcal{K}$. This is the super-analogue of the Cartan-Helgason theorem. We solve this problem via a rank one reduction and an understanding of reflections in singular roots, which generalize odd reflections in the theory of Kac-Moody superalgebras. An explicit presentation of spherical weights is provided for every pair when $\mathfrak{g}$ is indecomposable. |
| title | On the Cartan-Helgason theorem for supersymmetric pairs |
| topic | Representation Theory Mathematical Physics |
| url | https://arxiv.org/abs/2403.19145 |