Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution involution $σ$ in $Aut(G)$. Thus $σ$ is an automorphism of $G$ with $σ^2 = id$ and $H$ is an open subgroup of the set $G^\sigma=\{ g\in G: \sigma(g) = g\}.$

Now let $G/H$ be symmetric space then $T^*(G/H)\cong G^{\mathbb C}/H^{\mathbb C}$ is symmetric space?. Where $G^{\mathbb C}$ means complexification of lie group $G$