  Given any non-compact real simple Lie group G 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 G and on any
compact quotient $M=G/\Gamma$, with $\Gamma$ 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.

Differential Geometry