which will be used later.\
$X$ will always denote a smooth projective variety over a field $k$. Let $G$ be a reductive algebraic group defined over $k$ and let $\g$ denote its lie algebra. Fix a maximal torus $T
\subset G$ and a Borel $B$ containg $T$. Let $X_*(T)$ denote the set of 1-parameter subgroups of $T$ and let $X^*(T)$ be the character group of $T$. There exists a nondegenerate pairing, denoted $(\cdot , \cdot)$ : $X_{*}(T) \times
X^*(T)
\rightarrow \mathbb{Z}$. Let $\Phi \subset X^*(T)$ be the set of roots of $G$. Let $\Phi^+$ denote the set of positive roots corresponding to the choice of $B$ in $G$ and $\Delta $ = {$\alpha_1, \cdots ,\alpha_n \}$ a set of simple roots of $G$. Corresponding to this choice of simple roots, there exists a set of elements $\omega_i \in
X_*(T) \otimes {\mathbb Q}$ known as the fundamental weights with the property that $\langle
\omega_i,\alpha_j \rangle=\delta_{ij}$.\
For any root $\alpha$, there exists an isomorphism of $x_\alpha$ of $G_a$ with a closed subgroup $X_\alpha$ of $G$ with the property that $t \cdot x_\alpha(a) \cdot t^{-1} = x_\alpha(\alpha(t)a)$. $X_\alpha$ is known as the root group associated to $\alpha$.\
By a parabolic in $G$, we 