Let \({\mathcal L}\) be a real sub-Laplacian on \(\mathbb{R}^N\), \(N\geq 3\), and denote by \(G= (\mathbb{R}^N,0)\) its related homogeneous group. Let \(Q\) be the homogeneous dimension of \(G\). The main result is the following generalization of the classical Harnack inequality. Let \(Q/2 0\). A representation formula for functions \(u\) for which \({\mathcal L}u\) is a polynomial is also shown. As a consequence, some conditions are given ensuring that \(u\) is a polynomial whenever \({\mathcal L}u\) is a polynomial.