  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 note, 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.
