Fix an integer n and prime number p>>n. The compact groups

SL_n(Z_p) and SL_n(F_p[[t]]) are non-isomorphic, but, conjecturally, have isomorphic group algebra. I will talk about partial results related to this conjecture, and other connections between the representation theories of groups of the form G(O), where G is a fixed simple algebraic group, and O ranges over the collection of rings of integers of all local fields.