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Singular Integrals and the Principal Series, III.

A W Knapp1, E M Stein

  • 1Department of Mathematics, Cornell University, Ithaca, New York 14850.

Proceedings of the National Academy of Sciences of the United States of America
|November 1, 1974
PubMed
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This study simplifies calculations for unitary representations of real semisimple Lie groups. It introduces a new method using intertwining operators for nonunitary representations, reducing complex computations.

Area of Science:

  • Representation theory
  • Harmonic analysis
  • Lie group theory

Background:

  • Understanding unitary representations of real semisimple Lie groups is crucial in harmonic analysis.
  • Harish-Chandra's c-functions and Plancherel measure are fundamental but computationally intensive.
  • Intertwining operators play a key role in relating different representations.

Purpose of the Study:

  • To develop a streamlined method for constructing intertwining operators for unitary representations.
  • To reduce the computational complexity associated with Harish-Chandra's c-functions and the Plancherel measure.
  • To leverage nonunitary principal series representations to simplify calculations for unitary representations.

Main Methods:

  • Utilizing intertwining operators for nonunitary principal series representations.

Related Experiment Videos

  • Applying these operators to construct intertwining operators for all unitary representation series in the Plancherel formula.
  • Reducing computations involving Harish-Chandra's c-functions and Plancherel measure to simpler cases.

    • Main Results:

    • A novel framework for constructing intertwining operators for unitary representations of real semisimple Lie groups.
    • Significant reduction in computational complexity for key elements of the Plancherel formula.
    • Demonstration that computations can be performed using c-functions and measures from a minimal parabolic subgroup.

    Conclusions:

    • The developed machinery offers a more efficient approach to studying unitary representations.
    • This method simplifies the analysis of Plancherel formula components.
    • The findings provide a valuable tool for researchers in representation theory and harmonic analysis.