Hodge symmetry and decomposition on non-K\"ahler solvmanifolds

Let $G={\text{C}}^n{⋉}_{\varphi }{\text{C}}^m$ with a semi-simple action $\phi: \C^{n}\to GL_{m}(\C)$(notnecessarilyholomorphic)\mathrm{.}Suppose$G$hasalattice$\Gamma$\mathrm{.}Thenweshowthatinsomeconditionson$G$and$\Gamma$,$G/\Gamma$ 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-K"ahler Hermitian solvmanifolds satisfying the Hodge symmetry and decomposition.