On semisimple representations of universal lattices

Abstract

We study finite-dimensional semisimple complex representations of the universal lattices Γn,k = SLn(ℤ[x1, …, xk]) (n ≥ 3). One may obtain such a representation by specializing x1, …, xk to some complex values and composing the induced homomorphism Γn,k → SLn(ℂ) with a rational representation of SLn(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.