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Related Experiment Videos

Classes of unitarizable derived functor modules.

T J Enright1, R Parthasarathy, N R Wallach

  • 1Department of Mathematics, University of California at San Diego, La Jolla, CA 92093.

Proceedings of the National Academy of Sciences of the United States of America
|November 1, 1983
PubMed
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This study introduces a new method for constructing unitary representations of real semisimple Lie groups, addressing a long-standing challenge in representation theory. The findings prove unitarity for previously conjectured representations and extend beyond existing frameworks.

Area of Science:

  • Representation Theory
  • Harmonic Analysis
  • Mathematical Physics

Background:

  • The unitary dual of real semisimple Lie groups is a fundamental concept, yet its description remains elusive.
  • Constructing unitary representations is a key challenge, with standard methods like unitary induction from parabolic subgroups being insufficient.
  • Derived functor parabolic induction is necessary for all irreducible representations but lacks guaranteed unitarity.

Purpose of the Study:

  • To establish a framework where derived functor parabolic induction yields unitary representations for real semisimple Lie groups.
  • To prove unitarity for specific representations conjectured by Vogan and Zuckerman.
  • To extend unitarity proofs to representations outside the scope of existing conjectures.

Main Methods:

Related Experiment Videos

  • Utilizing derived functor parabolic induction within a specific setting.
  • Developing novel techniques to establish unitarity for induced representations.
  • Building upon established theories of Lie group representations.

Main Results:

  • A setting is described where derived functor parabolic induction guarantees unitary representations.
  • Unitarity is proven for a subset of representations previously conjectured by Vogan and Zuckerman.
  • Unitarity is demonstrated for representations not covered by prior conjectures.

Conclusions:

  • The developed framework provides a powerful tool for constructing and verifying unitary representations.
  • This work advances the understanding of the unitary dual for real semisimple Lie groups.
  • The results offer new insights into the behavior of derived functor parabolic induction.