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Branching problem of tensoring two Verma modules and its application to differential symmetry breaking operators

Reiji Murakami

Abstract

Kobayashi-Pevzner discovered in [Selecta Math., 2016] that the failure of the multiplicity-one property in the fusion rule of Verma modules of sl2 occurs exactly when the Rankin-Cohen bracket vanishes, and 1classified all the corresponding parameters. In this paper we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras g.

Branching problem of tensoring two Verma modules and its application to differential symmetry breaking operators

Abstract

Kobayashi-Pevzner discovered in [Selecta Math., 2016] that the failure of the multiplicity-one property in the fusion rule of Verma modules of sl2 occurs exactly when the Rankin-Cohen bracket vanishes, and 1classified all the corresponding parameters. In this paper we provide yet another characterization for these parameters, and give a precise description of indecomposable components of the tensor product. Furthermore, we discuss when the tensor products of two Verma modules are isomorphic to each other for semisimple Lie algebras g.
Paper Structure (12 sections, 9 theorems, 30 equations)

This paper contains 12 sections, 9 theorems, 30 equations.

Table of Contents

  1. Introduction.
  2. Proof of Theorem \ref{['generaltheorem']}.
  3. Preliminaries of parabolic Verma modules.
  4. Branching problem for parabolic Verma modules.
  5. Tensor products and Verma flags.
  6. Proof of Theorem \ref{['generaltheorem']}.
  7. Indecomposable components of $M(\mu')\otimes M(\mu")$ in the $\mathfrak{sl}_2$ case.
  8. Generic case.
  9. Singular case 1.
  10. Singular case 2.
  11. Proof of the equivalence (ii)$\Leftrightarrow$(iv) in Theorem \ref{['KPtheorem']}.
  12. Proof of Corollary \ref{['COR']}

Key Result

Theorem 1.1

Suppose that $\mu',\mu",\nu',\nu"\in \mathfrak{h}^{*}$ satisfy $\mu'+\mu"=\nu'+\nu"$. If each of the two sets $\{\mu',\mu" \}$ and $\{\nu',\nu" \}$ contains at least one anti-dominant element, then one has an isomorphism as $\mathfrak{g}$-modules:

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Example 1.7
  • Definition 2.1: Hum
  • Example 2.2
  • Definition 2.3: Kob
  • ...and 16 more