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Holomorphic Laplacian on the Lie ball and the Penrose transform

Hideko Sekiguchi

Abstract

We prove that any holomorphic function $f$ on the Lie ball of even dimension satisfying $Δf=0$ is obtained uniquely by the higher-dimensional Penrose transform of a Dolbeault cohomology for a twisted line bundle of a certain domain of the Grassmannian of isotropic subspaces. To overcome the difficulties arising from that the line bundle parameter is outside the {\it{good range}}, we use some techniques from algebraic representation theory.

Holomorphic Laplacian on the Lie ball and the Penrose transform

Abstract

We prove that any holomorphic function on the Lie ball of even dimension satisfying is obtained uniquely by the higher-dimensional Penrose transform of a Dolbeault cohomology for a twisted line bundle of a certain domain of the Grassmannian of isotropic subspaces. To overcome the difficulties arising from that the line bundle parameter is outside the {\it{good range}}, we use some techniques from algebraic representation theory.
Paper Structure (5 sections, 8 theorems, 26 equations)

This paper contains 5 sections, 8 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Penrose transform
  3. Differential operators on $G/K$
  4. Generalized Blattner formula
  5. Proof of Theorem \ref{['thm:main']} and Corollary \ref{['cor:230217']}

Key Result

Theorem 1.1

Let ${\mathcal{R}}$ be the cohomological integral transform (Penrose transform) defined in eqn:defPen below. Then ${\mathcal{R}}$ gives a topological $G$-isomorphism:

Theorems & Definitions (18)

  • Theorem 1.1: see Theorem \ref{['thm:main']}
  • Corollary 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2
  • Proposition 3.1
  • Remark 3.2
  • proof : Proof of Proposition \ref{['prop:3.1']}
  • Remark 3.3
  • Remark 3.4
  • ...and 8 more