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On the Cartan-Helgason theorem for supersymmetric pairs

Alexander Sherman

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.

On the Cartan-Helgason theorem for supersymmetric pairs

Abstract

Let be a supersymmetric pair arising from a finite-dimensional, symmetrizable Kac-Moody superalgebra . An important branching problem is to determine the finite-dimensional highest-weight -modules which admit a -coinvariant, and thus appear as functions in a corresponding supersymmetric space . 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 is indecomposable.
Paper Structure (33 sections, 30 theorems, 33 equations, 2 tables)

This paper contains 33 sections, 30 theorems, 33 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Questions in the super-setting
  3. Previously known results
  4. Main results
  5. Method of computation
  6. Singular reflections
  7. Consequences for simple spherical modules
  8. Supersymmetric spaces
  9. Outlook for the queer Kac-Moody setting
  10. Outline
  11. Acknowledgements
  12. Restricted root systems
  13. Supersymmetric pairs
  14. Restricted root system
  15. Nilpotent subalgebra $\mathfrak{n}_{\Sigma}$ and Iwasawa decomposition
  16. ...and 18 more sections

Key Result

Theorem 1.1

For the following supersymmetic pairs, we determine $P_{\Sigma}^+$ for every base $\Sigma\subseteq\Delta$, thus giving a full answer to question B: For the remaining supersymmetric pairs, which are we compute the spherical weights with respect to certain positive systems. Thus we obtain the answer to question B for certain positive systems.

Theorems & Definitions (73)

  • Theorem 1.1
  • Conjecture 1.2
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 63 more