On the Classification of Orbits of Symmetric Subgroups Acting on Flag Varieties of SL(2, k)

Abstract

Symmetric k-varieties are a generalization of symmetric spaces to general fields. Orbits of a minimal parabolic k-subgroup acting on a symmetric k-variety are essential in the study of symmetric k-varieties and their representations. In this article, we present the classification of these orbits for the group SL(2,k) for a number of base fields k, including finite fields and the 𝔭-adic numbers. We use the characterization in Helminck and Wang (1993), which requires one to first classify the orbits of the θ-stable maximal k-split tori under the action of the k-points of the fixed point group.

Key Words:

• k-split tori
• Minimal parabolic subgroups
• p-adic numbers
• Symmetric k-varieties
• Symmetric spaces

2000 Mathematics Subject Classification:

• 68Q40
• 20G15
• 20G20
• 22E15
• 22E46
• 43A85

ACKNOWLEDGMENT

The author Aloysius G. Helminck was partially supported by N.S.F. Grant DMS-0532140.

Notes

Communicated by K. C. Misra.