Commensurability and Representation Equivalent Arithmetic Lattices

Abstract

Prasad and 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. We observe that if we use the notion of representation equivalence of lattices, then Schanuel’s conjecture can be avoided. Further, the results are applicable in an S-arithmetic setting. We introduce a new relation “characteristic equivalence” on the class of arithmetic lattices, stronger than weak commensurability. This simplifies the arguments used in [11] to deduce commensurability type results.