On a theorem of Narasimhan and Ramanan on deformations

Abstract: Let $X$ be a smooth projective curve genus at least $3$, over an algebraically closed field $k$ of arbitrary characteristics. Let $\cM$ denote the moduli stack $\cM_X(\cH)$ of $\cH$-torsors on $X$, when $\cH$ {\em quasi-split absolutely simple, simply connected connected group scheme}. Using the theory of parahoric torsors and Hecke correspondences, we describe the cohomology groups $\text{H}^i(\cM, \cT_{{\cM}}), i = 0,1,2$ and $\text{H}^i(\cM, \Omega{_{\cM}}), i = 0,1,2$ in terms of the curve $X$. The classical results of Narasimhan and Ramanan are derived as a consequence.