On the space of $ K$-finite solutions to intertwining differential operators

Toshihisa Kubo*, Bent Orsted

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

Abstract

In this paper we give Peter-Weyl-type decomposition theorems for the space of K-finite solutions to intertwining differential operators between parabolically induced representations. Our results generalize a result of Kable for conformally invariant systems. The main idea is based on the duality theorem between intertwining differential operators and homomorphisms between generalized Verma modules. As an application we uniformly realize on the solution spaces of intertwining differential operators all small representations of (SL) over tilde (3,R) attached to the minimal nilpotent orbit.

Original languageEnglish
JournalRepresentation Theory
Volume23
Issue7
Pages (from-to)213-248
Number of pages36
ISSN1088-4165
DOIs
Publication statusPublished - Sept 2019

Keywords

  • CONFORMALLY INVARIANT-SYSTEMS
  • HEISENBERG ULTRAHYPERBOLIC EQUATION
  • Intertwining differential operators
  • K-finite solutions
  • MINIMAL REPRESENTATION
  • O(P
  • Peter-Weyl-type formulas
  • SINGULAR REPRESENTATION
  • SL(3
  • Torasso's representation
  • UNITARY REPRESENTATIONS
  • duality theorem
  • generalized Verma modules
  • hypergeometric polynomials
  • small representations

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