scholarly journals Quantum Unique Ergodicity on Locally Symmetric Spaces: the Degenerate Lift

2015 ◽  
Vol 58 (3) ◽  
pp. 632-650 ◽  
Author(s):  
Lior Silberman
Keyword(s):  
Symmetric Spaces ◽  
Differential Operators ◽  
Weak Limit ◽  
Iwasawa Decomposition ◽  
Probability Measures ◽  
Spectral Parameters ◽  
Locally Symmetric Spaces ◽  
Locally Symmetric Space ◽  
Connected Subgroup ◽  
Invariant Differential

AbstractGiven a measureon a locally symmetric spaceobtained as a weak-* limit of probability measures associated with eigenfunctions of the ring of invariant differential operators, we construct a measureon the homogeneous spaceX= Γ\Gthat liftsand is invariant by a connected subgroupA1⊂Aof positive dimension, whereG=NAKis an Iwasawa decomposition. If the functions are, in addition, eigenfunctions of the Hecke operators, thenis also the limit of measures associated with Hecke eigenfunctions on X. This generalizes results of the author with A.Venkatesh in the case where the spectral parameters stay away from the walls of the Weyl chamber.

scholarly journals GEODESIC TUBES ON LOCALLY SYMMETRIC SPACES

1993 ◽  
Vol 24 (4) ◽  
pp. 405-416
Author(s):  
B. J. PAPANTONIOU
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Locally Symmetric Spaces ◽  
Characteristic Relation ◽  
Locally Symmetric Space

In this paper we state and prove a characteristic relation which exists, between the eigenspaces of the Ricci transformation $R(N, - )N$ acting on the orthocomplement space of $N$ in $T_mM$ where $m \in M$, $M$ being a locally symmetric space, and the Weingarten map $S_N$ of small enough geodesic tubes of $M$.

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