# Understanding the local structure at a point on the moduli of semistable bundles of rank 2 of fixed determinant over a curve

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.