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On the algebraic construction and classification of Harish-Chandra modules
1Department of Mathematics, University of California, San Diego, La Jolla, California 92093.
Summary
A new functor maps Cartan subgroup representations to Harish-Chandra modules for real semisimple Lie groups. This method constructs modules and identifies conditions for irreducibility, aiding in classifying all such modules for linear Lie groups.
Area of Science:
- Representation theory
- Harmonic analysis
- Lie group theory
Background:
- Semisimple Lie groups are fundamental in mathematics and physics.
- Harish-Chandra modules are crucial for understanding infinite-dimensional representations.
- Cartan subgroups play a key role in the structure of Lie groups.
Purpose of the Study:
- To define a functor from representations of Cartan subgroups to Harish-Chandra modules.
- To provide an explicit construction for these modules.
- To classify irreducible Harish-Chandra modules for linear semisimple Lie groups.
Main Methods:
- Definition of a functorial mapping.
- Explicit module construction techniques.
- Development of criteria for module irreducibility.
Main Results:
- A functor is established between representation categories.
- Sufficient conditions for the irreducibility of constructed modules are provided.
- All irreducible Harish-Chandra modules are characterized for linear Lie groups.
Conclusions:
- The developed functor offers a systematic way to construct and study Harish-Chandra modules.
- This work advances the understanding of representation theory for semisimple Lie groups.
- The findings have implications for harmonic analysis on Lie groups.
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