Representation equivalence, characteristic equivalence and commensurability of arithmetic lattices

Gopal Prasad and A. S. Rapinchuk defined a notion of weakly
commensurable lattices in a semisimple group, and gave a
classification of weakly commensurable Zariski dense subgroups. A
motivation was to classify pairs of locally symmetric spaces
isospectral with respect to the Laplacian on functions. For this, in
higher ranks, they assume the validity of Schanuel's conjecture.
In this talk, we observe that if we use the stronger notion of
representation equivalence of lattices, then Schanuel's conjecture can
be avoided. Further, the results are also applicable in a
S-arithmetic setting.
We also introduce a new relation on the class of arithmetic lattices,
stronger than weak commensurability, which we call as characteristic
equivalence, and show that it simplifies some of the arguments used in
Prasad and Rapinchuk (2009) to deduce commensurability type results
from weak commensurability.