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 $G$
> 
> 
> 



Cotangent bundle of symmetric space is symmetric space?
The answer is yes. See this paper <http://arxiv.org/abs/0710.1543> (Introduction, and section 4.2 page 14). The basic ideas are that 


* if $M$ is a symmetric space, then $TM$ has a canonical structure of symmetric space (whatever the way you realize your symmetric space, apply the tangent functor to it);
* the duality of vector bundles extends to a duality of "symmetric bundles" as defined in the paper (i.e., vector bundles in the category of symmetric spaces): the dual vector bundle of a symmetric bundle inherits a structure of symmetric space which makes it a symmetric bundle.


Hence, $T^\*M$ inherits a canonical structure of symmetric space from that of $TM$.


