On a theorem of Narasimhan and Ramanan on deformations

Let $X$ be a smooth projective curve genus $G$ (as elaborated in \ref{main1}), over an algebraically closed field $k$ of arbitrary characteristics. Let $\text{cH}$ {\em be a tamely ramified absolutely simple, simply connected connected group scheme (see \eqref{quasisplitcase})}. Let $\text{cM}$ denote the moduli stack ${\text{cM}}_X(\text{cH})$ of $\text{cH}$-torsors on $X$ and ${\text{cM}}^{{}^s}$ be the open substack of {\em stable torsors}. Using the theory of parahoric torsors and Parahoric-correspondences, we describe the cohomology groups ${\text{H}}^i({\text{cM}}^{{}^s},{\text{cT}}_{{}_{\text{cM}}}),i=0,1,2$ and ${\text{H}}^i({\text{cM}}^{{}^s},{\mathrm{\Omega }}_{{}_{\text{cM}}}),i=0,1,2$ in terms of the curve $X$. The classical results of Narasimhan and Ramanan are derived as a consequence.