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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.
Summary
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:
- 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.
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