Hodge symmetry and decomposition on non-Kähler solvmanifolds

Let G=Cn⋉ϕCmG=Cn⋉ϕCm with a semi-simple action ϕ:Cn→GLm(C)ϕ:Cn→GLm(C) (not necessarily holomorphic). Suppose that GG has a lattice ΓΓ. Then we show that under some conditions on GG and ΓΓ, G/ΓG/Γ admits a Hermitian metric such that the space of harmonic forms satisfies the Hodge symmetry and decomposition. By this result we give many examples of non-Kähler Hermitian solvmanifolds satisfying the Hodge symmetry and decomposition.