Let $M$ and $N$ be dual lattices of rank $n$, i.e. $N\cong \text{Hom}(M, \Bbb{Z})$. Let $M_\Bbb{R}=M\otimes \Bbb{R}$ and similarly $N_\Bbb{R}$. Then $T^n\cong M_\Bbb{R}/M$ is the $n$-torus and we can also consider $T^*T^n\cong M_\Bbb{R}/M\times N_\Bbb{R}$. Let $\Sigma$ be a fan in $N_\Bbb{R}$ and for every cone $\sigma\in \Sigma$ we can define $$\sigma^\perp:=\{m\in M_\Bbb{R}| \langle m, s\rangle =0 \textrm{ for every }s\in\sigma\}$$ Let us call $p:M_\Bbb{R}\to T^n$ as the projection map and define $$L_\Sigma=\bigcup_{\sigma\in \Sigma}p(\sigma^\perp)\times (-\sigma)$$ Then $L_\Sigma$ is a Lagrangian in $T^*T^n$ and it defines a Legendrian $\Lambda_\Sigma$ in the cosphere bundle $T^\infty T^n=(T^*T^n\setminus \mathbf{0})/\Bbb{R}^+$.
Question: I am not quite sure how to prove they are indeed Lagrangian and Legendrian. What I have so far is that on $T^*T^n$ the Liouville form is $$\lambda_{[m_1], n_1}:T_{[m_1], n_1}(T^*T^n)\cong M_\Bbb{R}\times N_\Bbb{R}\to \Bbb{R} \textrm{ defined by }\lambda_{[m_1], n_1}(m_2, n_2)=\langle m_2, n_1\rangle$$ which seems to follow from the definition. But then if I choose the point in $L_\Sigma$ then $\langle m_1, n_1\rangle=0$ but I'm not sure what is $T_{[m_1], n_1}L_\Sigma$. Taking an example with $\Sigma=\sigma$ to be a ray in $N=\Bbb{Z}^2$ it seems like it should be $\sigma^\perp\times \text{span}(\sigma)$ but not sure what it should be in general. In that case the Liouville form vanishes as $\text{span}(\sigma)$ will be generated by $m_1$ over $\Bbb{R}$ and $m_2\in \text{span}(\sigma)$ and thus the subset defined is a Lagrangian. How to show this in the general case?
I am also not sure how one gets the Legendrian in the cosphere bundle, as far as I understand a Lagrangian gives us a Legendrian if it intersects the contact boundary transversely and the intersection is a Legendrian. But I don't know how to show this in this case as the cosphere bundle is a quotient and not identified with a subset unless we choose maybe a Riemannian metric. But not sure about this way of thinking so I would be grateful if someone helped me solve this problem. Thanks in advance.