Restriction Theorems and Root Systems for Symmetric Superspaces
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arXiv
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| Format: | Preprint |
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2024
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| author | Reif, Shifra Sahi, Siddhartha Serganova, Vera |
| author_facet | Reif, Shifra Sahi, Siddhartha Serganova, Vera |
| contents | In this paper we consider those involutions $θ$ of a finite-dimensional Kac-Moody Lie superalgebra $\mathfrak g$, with associated decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$, for which a Cartan subspace $\mathfrak a$ in $\mathfrak p_{\bar 0}$ is self-centralizing in $\mathfrak p$. For such $θ$ the restriction map $C_θ$ from $\mathfrak p$ to $\mathfrak a$ is injective on the algebra $P(\mathfrak p)^{\mathfrak k}$ of $\mathfrak k$-invariant polynomials on $\mathfrak p$. There are five infinite families and five exceptional cases of such involutions, and for each case we explicitly determine the structure of $P(\mathfrak p)^{\mathfrak k}$ by giving a complete set of generators for the image of $C_θ$. We also determine precisely when the restriction map $R_θ$ from $P(\mathfrak g)^{\mathfrak g}$ to $P(\mathfrak p)^{\mathfrak k}$ is surjective. Finally we introduce the notion of a generalized restricted root system, and show that in the present setting the $\mathfrak a$-roots $Δ(\mathfrak a,\mathfrak g)$ always form such a system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_04652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Restriction Theorems and Root Systems for Symmetric Superspaces Reif, Shifra Sahi, Siddhartha Serganova, Vera Representation Theory In this paper we consider those involutions $θ$ of a finite-dimensional Kac-Moody Lie superalgebra $\mathfrak g$, with associated decomposition $\mathfrak g=\mathfrak k\oplus\mathfrak p$, for which a Cartan subspace $\mathfrak a$ in $\mathfrak p_{\bar 0}$ is self-centralizing in $\mathfrak p$. For such $θ$ the restriction map $C_θ$ from $\mathfrak p$ to $\mathfrak a$ is injective on the algebra $P(\mathfrak p)^{\mathfrak k}$ of $\mathfrak k$-invariant polynomials on $\mathfrak p$. There are five infinite families and five exceptional cases of such involutions, and for each case we explicitly determine the structure of $P(\mathfrak p)^{\mathfrak k}$ by giving a complete set of generators for the image of $C_θ$. We also determine precisely when the restriction map $R_θ$ from $P(\mathfrak g)^{\mathfrak g}$ to $P(\mathfrak p)^{\mathfrak k}$ is surjective. Finally we introduce the notion of a generalized restricted root system, and show that in the present setting the $\mathfrak a$-roots $Δ(\mathfrak a,\mathfrak g)$ always form such a system. |
| title | Restriction Theorems and Root Systems for Symmetric Superspaces |
| topic | Representation Theory |
| url | https://arxiv.org/abs/2401.04652 |