Real semisimple Lie groups and balanced metrics

Abstract

Given any non-compact real simple Lie group ${\mathrm{G}}_o$ of inner type and even dimension, we prove the existence of an invariant complex structure $\mathrm{J}$ and a Hermitian balanced metric on ${\mathrm{G}}_o$ and on any compact quotient $\mathrm{M}=\Gamma \backslash {\mathrm{G}}_o$, with $\Gamma$ a cocompact lattice. We also prove that $(\mathrm{M},\mathrm{J})$ does not carry any pluriclosed metric, in contrast to the case of even dimensional compact Lie groups, which admit pluriclosed but not balanced metrics.