On semisimple representations of universal lattices. (Q1049917)

Finite-dimensional semisimple complex representations of the universal lattices \(\Gamma_{n,k}=\text{SL}_n(\mathbb{Z}[x_1,\dots,x_k])\) (\(n\geq 3\)) are studied. One may obtain such a representation by specializing \(x_1,\dots,x_k\) to some complex values and composing the induced homomorphism \(\Gamma_{n,k}\to\text{SL}_n(\mathbb{C})\) with a rational representation of \(\text{SL}_n(\mathbb{C})\). It is shown that every semisimple representation coincides, on a subgroup of finite index, with a direct sum of tensor products of representations obtained in this way.