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Related Experiment Videos

Singular integrals and the principal series.

A W Knapp1, E M Stein

  • 1SCHOOL OF MATHEMATICS, INSTITUTE FOR ADVANCED STUDY, PRINCETON, NEW JERSEY.

Proceedings of the National Academy of Sciences of the United States of America
|June 1, 1969
PubMed
Summary

This study extends the L(2) theory of singular integral operators to nilpotent Lie groups, a key step in understanding group representations. Researchers identified specific intertwining operators, determining irreducible representations of principal series in simple Lie groups.

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Area of Science:

  • Harmonic analysis
  • Representation theory
  • Functional analysis

Background:

  • The L(2) theory of singular integral operators is well-established for Euclidean space (IRn).
  • Nilpotent Lie groups present a more complex setting for analyzing these operators.
  • Understanding representations of Lie groups is fundamental in various areas of mathematics and physics.

Purpose of the Study:

  • To extend the L(2) theory of singular integral operators to the domain of nilpotent Lie groups.
  • To investigate the nature of intertwining operators for representations of simple Lie groups of real rank one.
  • To classify irreducible representations within the principal series of these Lie groups.

Main Methods:

  • Utilizing the framework of L(2) theory for singular integral operators.

Related Experiment Videos

  • Analyzing intertwining operators associated with group representations.
  • Applying techniques from the theory of Lie groups and their representations.

    • Main Results:

    • The study successfully extends the L(2) theory of singular integral operators to nilpotent Lie groups.
    • Intertwining operators for representations of simple Lie groups of real rank one are identified as belonging to this extended theory.
    • A complete determination of which representations of the principal series of these groups are irreducible has been achieved.

    Conclusions:

    • The generalization of L(2) theory to nilpotent Lie groups provides a powerful tool for representation analysis.
    • The characterization of intertwining operators offers new insights into the structure of representations.
    • This work contributes to the classification and understanding of irreducible representations in Lie group theory.