Real semisimple Lie groups and balanced metrics

Given any non-compact real simple Lie group Go of inner type and even dimension, we prove the existence of an invariant complex structure J and a Hermitian balanced metric with vanishing Chern scalar curvature on Go and on any compact quotient M = Go/Γ, with Γ a cocompact lattice. We also prove that (M, J) does not carry any pluriclosed metric, in contrast to the case of even dimensional compact Lie groups, which admit pluriclosed but not balanced metrics.