Let $X$ be a curve of genus $g$.
If $d>n(2g-1)$,then for any vector bundle $E$ of rank $n$ and degree $d$ over $X$ has $H^1(X,E)=0$ and $E$ is generated by its global sections. For such a vector bundle $dim(H^0(X,E)=d+n(1-g)$.
Consider $R(2,p):=$ set of all holomorphic maps from $h:X\rightarrow G(2,p)$ such that
- $h^{*}(T):=E(h)$ has degree $d$ with determinant $\mathcal{O}_X$ where $T$ is the tautological bundle over $G(2,p)$
- The map on sections $\mathbb{C}^p\rightarrow H^{0}(M,E(h))$ induced from the quotient bundle map $\mathbb{C}^p\times X\rightarrow E(h)$ is an isomorphism.
$R(2,d)$ is a non-singular quasi-projective variety. $Sl(p)$ acts naturally on $R(2,d)$. The geometric invariant theory quotient $R(2,d)//Sl(p)$ can be naturally identified with $M(2,d)$.
Where, $M(2,d):=\frac{\text{set of all vector bundles of rank 2 and degree d with determinant} \mathcal{O}_X}{\sim}$
$\sim$ is the equivalence defined by $E_1\sim E_2$ if and only if $gr(E_1)\sim gr(E_2)$
$M(2,d)$ has a decomposition $M(2,d)=\mathbb{Z}^{2g}\sqcup(\mathbb{k}-\mathbb{Z}^{2g})\sqcup (\text{points corresponding to stable vector bundles})$, where $\mathbb{Z}^{2g}$ represents all non-stable semistable vector bundles which is equivalent some $L\oplus L$ and $\mathbb{k}-\mathbb{Z}^{2g}$ represents all non-stable semistable points which are equivalent to $L\oplus L^{-1}$ such that $L\ncong L^{-1}$
Consider a point $l=[L\oplus L^{-1}]\in \mathbb{Z}^{2g}\subset M(2,d)$.
Let $W_l$ be the etale slice (by Luna's theorem) of the unique closed orbit in $R(2,d)^{ss}$ over $l$.
By deformation theory, the normal space of the orbit at a point $l$ which represents $[L+L^{-1}]$, is
$N_l=H^{1}(End_0(L\oplus L))\cong H^1(\mathcal{O})\otimes sl(2)$, where the subscript $0$ denotes the trace free-part.
Can someone please explain the last line? Why is the normal space to the orbit over a point is $H^{1}(End_0(L\oplus L^{-1}))$?