Journal of Modern Dynamics (JMD)

Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

Pages: 335 - 357, Issue 3, July 2009      doi:10.3934/jmd.2009.3.335

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Felipe A. Ramírez - Department of Mathematics, University of Michigan, Ann Arbor, MI 48104, United States (email)

Abstract: We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra - one nilpotent and the other semisimple. Second, we consider actions arising from two commuting unipotent flows that come from an embedded copy of $\overline{\SL(2,\RR)}^{k} \times \overline{\SL(2,\RR)}^{l}$. In both cases we show that any smooth $\RR$-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function. These results build on theorems of D. Mieczkowski, where the same is shown for actions on $(\SL(2,\RR) \times \SL(2,\RR))$/Γ.

Keywords:  cohomology, smooth cocycle, representation theory.
Mathematics Subject Classification:  Primary: 37C85, 37D30, 37A20; Secondary: 22E46, 22F30.

Received: October 2008;      Revised: March 2009;      Available Online: August 2009.