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Indefinite intertwining operators.

M W Baldoni-Silva1, A W Knapp

  • 1Dipartimento di Matematica, Università degli Studi di Trento, 38050 Povo (TN), Italy.

Proceedings of the National Academy of Sciences of the United States of America
|February 1, 1984
PubMed
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This study provides formulas for Langlands parameters of unitary representations in semisimple Lie groups, linking unitarity to root conditions and confirming related conjectures.

Area of Science:

  • Representation Theory
  • Harmonic Analysis
  • Algebraic Groups

Background:

  • The study of irreducible unitary representations is central to harmonic analysis and representation theory.
  • Langlands parameters provide a crucial link between number theory and the representation theory of algebraic groups.
  • Maximal parabolic subgroups are fundamental in the structure theory of semisimple Lie groups.

Purpose of the Study:

  • To derive formulas that bound the Langlands parameters for irreducible unitary representations.
  • To investigate the relationship between the unitarity of representations and specific conditions on roots within Lie groups.
  • To provide evidence for the sharpness of these derived formulas and confirm existing conjectures.

Main Methods:

  • Analysis of linear connected semisimple Lie groups.

Related Experiment Videos

  • Formulas are derived for representations obtained from maximal parabolic subgroups.
  • Examination of conditions on roots related to the unitarity of representations.

    • Main Results:

    • Formulas are established that limit the Langlands parameters for a broad class of semisimple Lie groups.
    • A direct correlation is shown between the unitarity of representations and the number of roots satisfying certain criteria.
    • Evidence suggests these formulas are precise and not overly restrictive.

    Conclusions:

    • The derived formulas offer new insights into the structure of unitary representations.
    • The findings support and confirm aspects of conjectures linking unitary Langlands parameters with cohomological induction.
    • This work advances the understanding of representation theory for semisimple Lie groups.