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epsilon factor of representations of classical groups
I Piatetski-Shapiro1, S Rallis
1Department of Mathematics, Yale University, New Haven, CT 06520.
Summary
This study introduces L functions and epsilon factors for isometry groups of bilinear forms. It develops theories for local and global zeta integrals, determining correct epsilon factors for representations.
Area of Science:
- Algebraic groups
- Representation theory
- Number theory
Background:
- The study of L functions and epsilon factors is crucial in number theory and representation theory.
- Isometry groups of bilinear forms, denoted G = U(V, h), play a significant role in various mathematical fields.
- Understanding the properties of these groups and their representations is essential for deeper mathematical insights.
Purpose of the Study:
- To develop a theory of L functions and epsilon factors for the natural representation (r) of the identity component of G (denoted G(o)).
- To establish a framework for local and global zeta integrals associated with these isometry groups.
- To determine the correct epsilon factors for local representations and discuss the issue of local L factors.
Main Methods:
- Utilizing special intertwining operators with careful analysis of their normalization.
- Developing a theory of local and global zeta integrals for the isometry group G.
- Applying these techniques to determine epsilon factors for local representations.
Main Results:
- A comprehensive theory of L functions and epsilon factors for the natural representation r of G(o) has been developed.
- The theory of local and global zeta integrals for G has been established.
- The correct epsilon factors associated with local representations have been determined.
Conclusions:
- The developed theory provides a robust framework for studying L functions and epsilon factors in the context of isometry groups.
- The precise determination of epsilon factors advances the understanding of local representations.
- Further discussion on the correct local L factor is warranted for complete characterization.
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