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Hodge symmetry and decomposition on non-Kähler solvmanifolds

Let $G=\C^{n}\ltimes_ϕ \C^{m}$ with a semi-simple action $ϕ: \C^{n}\to GL_{m}(\C)$ (not necessarily holomorphic). Suppose $G$ has a lattice $Γ$. Then we show that in some conditions on $G$ and $Γ$, $G/Γ$ 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.