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Journal of Lie Theory 22 (2012), No. 2, 437--463
Copyright Heldermann Verlag 2012

Invariant Orders on Hermitian Lie Groups

Gabi Ben Simon
Departement Mathematik, ETH Zürich, Rämistr. 101, 8092 Zürich, Switzerland

Tobias Hartnick
Mathematics Department, Technion, Haifa 3200, Israel

We study three natural bi-invariant partial orders on a certain covering group of the automorphism group of a bounded symmetric domain of tube type; these orderings are defined using the geometry of the Shilov boundary, Lie semigroup theory and quasimorphisms respectively. Our main result shows that these orders are related by two inclusion relations. In the case of SL2(R), where R stands for the real numbers, we can show that they coincide. We also prove a related coincidence of orders for the universal covering of the group of homeomorphisms of the circle.

Keywords: Hermitian Lie groups, invariant orders, quasimorphisms, Lie semigroups, bounded cohomology.

MSC: 06A06, 06F15, 11E57, 22E46, 51L99

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