On semisimple representations of universal lattices

Abstract

We study finite-dimensional semisimple complex representations of the universal lattices ${\Gamma }_{n,k}={\mathrm{SL}}_n(Z[x_1,\dots ,x_k])$ ($n\ge 3$). One may obtain such a representation by specializing $x_1,\dots ,x_k$ to some complex values and composing the induced homomorphism ${\Gamma }_{n,k}\to {\mathrm{SL}}_n(C)$ with a rational representation of ${\mathrm{SL}}_n(C)$. We show that any semisimple representation coincides, on a subgroup of finite index, with a direct sum of tensor products of representations obtained in this way.