On the Cartan-Helgason theorem for supersymmetric pairs

Alexander Sherman

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On the Cartan-Helgason theorem for supersymmetric pairs

Alexander Sherman

Representation Theory · Submitted 28 Mar 2024

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim An explicit presentation of spherical weights is provided for every pair when $\mathfrak{g}$ is indecomposable.

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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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $(\mathfrak{g},\mathfrak{k})$ be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac-Moody superalgebra $\mathfrak{g}$.	background 0.99	0.09	0.02	asserted 0.91
s1	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 $\mathcal{G}/\mathcal{K}$.	background 1.00	0.06	0.05	asserted 1.00
s2	This is the super-analogue of the Cartan-Helgason theorem.	background 0.75	0.09	0.07	asserted 1.00
s3	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.	method 0.98	0.94	0.36	asserted 1.00
s4	An explicit presentation of spherical weights is provided for every pair when $\mathfrak{g}$ is indecomposable.	result 0.81	0.95	0.59	asserted 1.00

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Original abstract

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.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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They also have important connections to interpolation polynomials (see [ref] , [ref] , and [ref] ), integrable systems (see [ref] ), and physics (see [ref] and [ref] ).

Eigenvalues of supersymmetric Shimura operators and interpolation polynomials · Supersymmetric spaces · 0.96

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