Polarized relations on horizontal SL(2)s

TL;DR

This paper introduces a new relation on SL(2)-orbits in Mumford-Tate domains, generalizing the SL(2)-orbit theorem, and provides an algorithm for computing it, with applications to various types of domains.

Contribution

It defines a polarized relation on SL(2)-orbits in Mumford-Tate domains and develops an algorithm to compute this relation, extending the classical SL(2)-orbit theorem.

Findings

Defined a relation compatible with partial orders on nilpotent and boundary orbits.

Developed an algorithm for computing the relation.

Applied the framework to period, Hermitian symmetric, and flag domains.

Abstract

We introduce a relation on real conjugacy classes of SL(2)-orbits in a Mumford-Tate domain D which is compatible with natural partial orders on the sets of nilpotent orbits in the corresponding Lie algebra and boundary orbits in the compact dual. A generalization of the SL(2)-orbit theorem to such domains leads to an algorithm for computing this relation, which is worked out in several examples and special cases including period domains, Hermitian symmetric domains, and complete flag domains, and used to define a poset of equivalence classes of multivariable nilpotent orbits on D.