Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces

Toshihisa Kubo

Abstract

In this paper we classify and construct differential symmetry breaking operators $\mathbb{D}$ from a line bundle over the real projective space $\mathbb{R}\mathbb{P}^n$ to a vector bundle over $\mathbb{R}\mathbb{P}^{n-1}$. We further determine the factorization identities of $\mathbb{D}$ and the branching laws of the corresponding generalized Verma modules of $\mathfrak{sl}(n+1,\mathbb{C})$. By utilizing the factorization identities, the $SL(n,\mathbb{R})$-representations realized on the image $\text{Im}(\mathbb{D})$ are also investigated.

Differential symmetry breaking operators from a line bundle to a vector bundle over real projective spaces

Abstract

In this paper we classify and construct differential symmetry breaking operators from a line bundle over the real projective space to a vector bundle over . We further determine the factorization identities of and the branching laws of the corresponding generalized Verma modules of . By utilizing the factorization identities, the -representations realized on the image are also investigated.
Paper Structure (56 sections, 50 theorems, 339 equations)

This paper contains 56 sections, 50 theorems, 339 equations.

Table of Contents

  1. Introduction
  2. Differential symmetry breaking operators and main problems
  3. Classification and construction of $\mathbb{D}$
  4. Factorization identity of $\mathbb{D}$
  5. The image $\mathrm{Im}(\mathbb{D})$
  6. $SL$ vs $GL$
  7. Classification of SBOs $\mathbb{A}$ between line bundles over $\mathbb{R}\mathbb{P}^n$ and $\mathbb{R}\mathbb{P}^{n-1}$
  8. F-method
  9. Branching law of generalized Verma modlues
  10. Organization of the paper
  11. Preliminaries: the F-method
  12. Notation
  13. Duality theorem
  14. Algebraic Fourier transform $\widehat{\;\;\cdot\;\;}$ of Weyl algebras
  15. Fourier transformed representation $\widehat{d\pi_{(\sigma,\lambda)^*}}$
  16. ...and 41 more sections

Key Result

Theorem 2.3

There is a natural linear isomorphism Equivalently, Here, for $\varphi \in \operatorname{Hom}_{P'}(W^\vee, M^{{\mathfrak g}}_{{\mathfrak p}}(V^\vee))$ and $F \in C^\infty(G/P,\mathcal{V})\simeq C^\infty(G,V)^P$, the section $\EuScript{D}_{H\to D}(\varphi)F \in C^\infty(G'/P',\mathcal{W})\simeq C^\infty(G',W)^{P'}$ is given by where $\varphi(w^\vee)=\sum_j u_j\otimes v_j^\vee \in M^{{\mathfrak g

Theorems & Definitions (95)

  • Theorem 2.3: Duality theorem
  • Theorem 2.12: KP1
  • Theorem 2.19: F-method, KP1
  • Theorem 2.20: KP1
  • Remark 3.7
  • Remark 3.12
  • Theorem 4.10
  • Theorem 4.11
  • Theorem 4.15
  • Theorem 4.16
  • ...and 85 more