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First, let me describe the premise. Let $X$ be a smooth projective curve of genus greater than 2 over $\mathbb{C}$, let $x,y\in X$ two closed points, and let $M_{y-x}$ denote the moduli space of S-equivalence classes of rank 2 semistable vector bundles on $X$ of fixed determinant $\mathcal{O}(y-x).$ $M_{y-x}$ is constructed as the GIT quotient $\mathcal{R}//SL(p),$ where $p$ is a large even number, and $\mathcal{R}$ is a smooth open subvariety of the Grassmannian of 2-dimensional quotients of $\mathbb{C}^p$. There exists a surjective map $\mathcal{R}^{ss}\rightarrow M_{y-x}$, where $\mathcal{R}^{ss}$ denotes the semistable points under the $SL(p)$-action. Let $l$ be a strictly semistable (i.e. semistable but not stable) bundle. Using Jordan-Holder filtration, it is easy to see that the S-equivalence class of $l$ is of the form $[l] = [L\oplus L^{-1}(y-x)]$. We know that There is a unique closed orbit lying over $[l]$ in $\mathcal{R}^{ss}$.

In a paper that I'm reading, the following has been claimed:

for each point $h$ on this closed orbit, the normal space of this orbit at $h$ is isomorphic to $\mathcal{N}=H^1(X,End_0(L\oplus L^{-1}(y-x)).$ Moreover, the stabilizer of the point $h$ is isomorphic to $\mathbb{C}^{*}$.

My question is: Does the above follow from general theory of GIT? Can someone give some idea on why it should be true? any reference would be highly appreciated.

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