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Holographic operators for the tensor products of the spaces of holomorphic functions on Hermitian symmetric spaces of tube type

Ryosuke Nakahama

Abstract

We consider a tensor product of two spaces of holomorphic functions on a Hermitian symmetric space of tube type. Then generically this is decomposed into a direct sum of irreducible subrepresentations. In this manuscript, we construct the intertwining operator (holographic operator) from each irreducible summand to the tensor product as an integral operator. This gives a generalization of the result by Kobayashi--Pevzner (2016).

Holographic operators for the tensor products of the spaces of holomorphic functions on Hermitian symmetric spaces of tube type

Abstract

We consider a tensor product of two spaces of holomorphic functions on a Hermitian symmetric space of tube type. Then generically this is decomposed into a direct sum of irreducible subrepresentations. In this manuscript, we construct the intertwining operator (holographic operator) from each irreducible summand to the tensor product as an integral operator. This gives a generalization of the result by Kobayashi--Pevzner (2016).
Paper Structure (15 sections, 27 theorems, 183 equations)

This paper contains 15 sections, 27 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Jordan algebras and Kantor--Koecher--Tits construction
  4. Jordan frames and restricted root systems
  5. Representations on the spaces of holomorphic functions
  6. Tensor product of the spaces of holomorphic functions
  7. Definition of contour of integration
  8. Construction of integral holographic operators
  9. Main theorems
  10. Proof of single-valuedness of the integrand
  11. Proof of continuity in $L^\infty_{\mathrm{loc}}$
  12. Proof of intertwining property on $L^\infty_{\mathrm{loc}}$
  13. Proof of holomorphy
  14. Proof of Theorem \ref{['main_thm2']}
  15. Computation of images of minimal $\widetilde{K}$-types for scalar-valued case

Key Result

Theorem 1.1

Let $\lambda,\mu\in\mathbb{C}$, $l\in\mathbb{Z}_{\ge 0}$. Then the map intertwines the $\widetilde{SL}(2,\mathbb{R})$-action.

Theorems & Definitions (45)

  • Theorem 1.1: Rankin--Cohen CR
  • Theorem 1.2: Kobayashi--Pevzner KP3
  • Theorem 2.1: Kobayashi, Kmf1
  • Theorem 2.2: Kobayashi, Kmf1
  • Theorem 2.3: N1
  • Theorem 2.4: N1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 35 more