Semisimple representations of universal lattices

The groups G(n,k) = SL(n, Z[x_1,…,x_k]) were named “universal
lattices” by Y. Shalom. They share some interesting properties with the
groups SL(n,O) where O is the ring of integers of a number field, e.g .
Kazhdan’s property T.
In this work we study finite dimensional semisimple complex representations
of G=G(n,k). A natural way of producing such representations is by
specializing x_1,…,x_k to some complex values, and compose the induced
homomorphism G–>SL(n,C) with a representation of SL(n,C).
We prove that any semisimple representation of G coincides on a finite index
subgroup with a tensor product of representations obtained in this way. The
proof is based on the method employed by Bass, Milnor and Serre to prove
superrigidity of SL(n,O) using the congruence subgroup property.