scholarly journals Real Hardy spaces on real rank 1 semisimple Lie groups

2005 ◽  
Vol 31 (2) ◽  
pp. 281-343 ◽  
Author(s):  
Takeshi KAWAZOE
Keyword(s):  
Lie Groups ◽  
Hardy Spaces ◽  
Real Rank ◽  
Semisimple Lie Groups

Whittaker Functions on Real Semisimple Lie Groups of Rank Two

2010 ◽  
Vol 62 (3) ◽  
pp. 563-581 ◽  
Author(s):  
Taku Ishii
Keyword(s):  
Lie Groups ◽  
Principal Series ◽  
Real Rank ◽  
Whittaker Functions ◽  
Semisimple Lie Groups ◽  
Explicit Formulas ◽  
Series Representations ◽  
Principal Series Representations

AbstractWe give explicit formulas forWhittaker functions on real semisimple Lie groups of real rank two belonging to the class one principal series representations. By using these formulas we compute certain archimedean zeta integrals.

Rigidity of weakly hyperbolic actions of higher real rank semisimple Lie groups and their lattices

2001 ◽  
Vol 21 (1) ◽  
pp. 121-164 ◽  
Author(s):  
GREGORY A. MARGULIS ◽  
NANTIAN QIAN
Keyword(s):  
Lie Groups ◽  
Left Translation ◽  
Real Rank ◽  
Semisimple Lie Groups ◽  
Rigidity Result ◽  
Global Rigidity ◽  
Local Rigidity ◽  
Algebraic Actions ◽  
Anosov Actions ◽  
Higher Rank

Under some weak hyperbolicity conditions, we establish C^0- and C^\infty-local rigidity theorems for two classes of standard algebraic actions: (1) left translation actions of higher real rank semisimple Lie groups and their lattices on quotients of Lie groups by uniform lattices; (2) higher rank lattice actions on nilmanifolds by affine diffeomorphisms. The proof relies on an observation that local rigidity of the standard actions is a consequence of the local rigidity of some constant cocycles. The C^0-local rigidity for weakly hyperbolic standard actions follows from a cocycle C^0-local rigidity result proved in the paper. The main ingredients in the proof of the latter are Zimmer's cocycle superrigidity theorem and stability properties of partially hyperbolic vector bundle maps. The C^\infty-local rigidity is deduced from the C^0-local rigidity following a procedure outlined by Katok and Spatzier.Using similar considerations, we also establish C^0-global rigidity of volume preserving, higher rank lattice Anosov actions on nilmanifolds with a finite orbit.

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