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Let $G=\C^{n}\ltimes_{\phi} \C^{m}$ with a semi-simple action $\phi:\C^{n}\to GL_{m}(\C)$ (not necessarily holomorphic). Suppose $G$ has a lattice$\Gamma$. Then we show that in some conditions on $G$ and $\Gamma$, $G/\Gamma$admits a Hermitian metric such that the space of harmonic forms satisfies theHodge symmetry and decomposition. By this result we give many examples ofnon-K\"ahler Hermitian solvmanifolds satisfying the Hodge symmetry anddecomposition.

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