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The notion of $\Gamma$-symmetric space is a natural generalization of theclassical notion of symmetric space based on $\z_2$-grading of Lie algebras. Inour case, we consider homogeneous spaces $G/H$ such that the Lie algebra $\g$of $G$ admits a $\Gamma$-grading where $\Gamma$ is a finite abelian group. Inthis work we study Riemannian metrics and Lorentzian metrics on the Heisenberggroup $\mathbb{H}_3$ adapted to the symmetries of a $\Gamma$-symmetricstructure on $\mathbb{H}_3$. We prove that the classification of$\z_2^2$-symmetric Riemannian and Lorentzian metrics on $\mathbb{H}_3$corresponds to the classification of left invariant Riemannian and Lorentzianmetrics, up to isometries. This gives examples of non-symmetric Lorentzianhomogeneous spaces.

Pseudo-Riemannian Symmetries on Heisenberg group $\mathbb{H}_{3}$