On Representations of Classical Groups Over Principal Ideal Local Rings of Length Two

Abstract

We study the complex irreducible representations of special linear, symplectic, orthogonal, and unitary groups over principal ideal local rings of length two. We construct a canonical correspondence between the irreducible representations of all such groups that preserves dimensions. The case for general linear groups has already been proved by the author.

Key Words:

• Clifford theory
• Principal ideal local ring
• Representations of classical groups
• Representations of general linear groups

2000 Mathematics Subject Classification:

• Primary 20G25
• Secondary 20C15

ACKNOWLEDGMENTS

The author is greatly thankful to Uri Onn for suggesting these questions and for many useful comments on a preliminary version of this article. It is a great pleasure to thank Amritanshu Prasad for very useful feedback. The author also thanks the referee for helpful suggestions concerning the presentation of the article. The author was partially supported by Center for Advanced Studies in Mathematics at Ben Gurion University, Beer Sheva, Israel.

Notes

Communicated by M. Cohen.