I am trying to understand Klyachko's classification of toric vector bundles on a toric variety ( his article: Equivariant vector bundles on toric varieties and some problems of linear algebra). I am reading Sam Payne's version of it as presented in the preliminaries of his article Moduli of Toric Vector Bundles. I am having some troubles understanding two things. The setting:
Let $N$ be a lattice, $M = \text{Hom}(N,\mathbb{Z})$ the dual lattice, $\Delta$ a fan in $N \otimes_\mathbb{Z}\mathbb{R}$, $ X = X(\Delta)$ the corresponding toric variety and $\mathscr{E}$ a toric vector bundle on $X$. An algebraic action of the torus $T$ on $\Gamma(X,\mathscr{E})$, the vector space of global sections, is given by $$ (t\cdot s)(x) = t(s(t^{-1}x))$$ for $t\in T, s\in\Gamma(X,\mathscr{E})$ and $x\in X$. So far so good.
This action induces the following decomposition into T-eigenspaces $$ \Gamma(X,\mathscr{E}) = \oplus_{u\in M}\Gamma(X,\mathscr{E})_u, $$ where $$ t\cdot s = \chi^{u}(t)s$$ for $t\in T$ and $s\in \Gamma(X,\mathscr{E})_u$.
Question 1: Why does this decomposition exist? I have read (Cox, Toric Varieties Porp 1.1.2) that, for an algebraic action, such a decomposition exists if the action is linear and the vector space is of a finite dimension. I see that the above action is linear but cannot see why the vector space $\Gamma(X,\mathscr{E})$ is finite dimensional.
Question 2: I know that for $u\in M$ the character $\chi^{u}$ is a regular function on $X$. I don't understand why in the above setting, the eigenfuction $\chi^{u}$ is an element of $\Gamma(X, O_X)_{-u}$
Additional questions (after the above were answered thanks to sti9111) Continuing my pursue of understanding Klyachko's theorem, I am trying to understand the following point mentioned both in Payne (p.7) and in Gonzalez (p. 7). Continuing with Payne's notation let $E$ be the fiber at the identity element of the Torus, $x_0$. Consider the map $\Gamma(X,\mathscr{E})_u\to E$ given by the evaluation at $x_0$. For $\sigma\in\Delta$, denote with $E^{\sigma}_{u}$ the image of $\Gamma(U_\sigma,\mathscr{E})_u$ under this evaluation map. For a ray $\rho\in\Delta$, we denote with $E^{\rho}(i)=E^{\rho}_{u}$ for any $u$ such that $<u,v_\rho>=i$.
My question: both in Payne's proof of Klyachko's classification theorem (p.7) and in Gonzalez's article (p.7), the fllowing is mentioned $E^{\sigma}_{u}=\cap_{\substack{\rho\preceq\sigma}}E^{\rho}(<u,v_\rho>)$ and in Gonzalez, the image of $H^{0}(X,\mathscr{E})_u$ under the evaluation map is equal to $\mathscr{E}^{\rho1}(<u,v_1>)\cap\dots\cap\mathscr{E}^{\rho_{d}}(<u,v_d>)$. I cannot see why this holds.
Explain the Toric vector bundle - Klyachko's decomposition here could be a little bit difficult, however, I think that the paper "Okounkov bodies on projectivizations of rank two toric vector bundles" by Jose Luis Gonzalez, on pages 6 and 7 gives a good explanation.