Proper SL(2,R)-actions on homogeneous spaces
Maciej Bochenski, Piotr Jastrzebski, Takayuki Okuda, Aleksy Tralle
TL;DR
This paper investigates conditions under which SL(2,R) can act properly on certain homogeneous spaces, providing criteria based on Lie algebra structures and illustrating with examples.
Contribution
It establishes a new criterion for proper SL(2,R)-actions on homogeneous spaces of reductive type, expanding understanding of group actions in geometric contexts.
Findings
A criterion involving maximally split abelian subspaces for proper actions.
Examples of homogeneous spaces admitting proper SL(2,R)-actions.
Application of Kobayashi's properness criterion to new cases.
Abstract
We study the existence problem of proper actions of SL(2,R) on homogeneous spaces G/H of reductive type. Based on Kobayashi's properness criterion [Math. Ann. (1989)], we show that G/H admits a proper SL(2,R)-action via G if a maximally split abelian subspace of Lie H is included in the wall defined by a restricted root of Lie G. We also give a number of examples of such G/H.
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