Equal rank local theta correspondence as a strong Morita equivalence

• Bram Mesland ^[1] ; Mehmet Haluk Sengün ^[2]
  1. [1] Leiden University

     Leiden University

     Países Bajos

     
  2. [2] University of Sheffield

     University of Sheffield

     Reino Unido

     
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 4, 2024, págs. 1-43
• Idioma: inglés
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• Resumen

  • Let (G, H) be one of the equal rank reductive dual pairs (Mp2n, O2n+1) or (Un, Un) over a nonarchimedean local field of characteristic zero. It is well-known that the theta correspondence establishes a bijection between certain subsets, say G θ and H θ , of the tempered duals of G and H. We prove that this bijection arises from an equivalence between the categories of representations of two C∗-algebras whose spectra are G θ and H θ . This equivalence is implemented by the induction functor associated to a Morita equivalence bimodule (in the sense of Rieffel) which we construct using the oscillator representation. As an immediate corollary, we deduce that the bijection is functorial and continuous with respect to weak inclusion. We derive further consequences regarding the transfer of characters and preservation of formal degrees