notion of connection on Torsors Hi,
   Can anyone please enlighten me on the notion of connection for G-torsors?
Edit (by WW) For some reason the OP decides to add clarification below as an answer rather than editing the question. I'm copying the clarification he/she provided:
I have a smooth complex projective curve $X$ and $G$ is semisimple, simply connected complex algebraic group. The paper I was trying to read associates a group $G$ scheme over $X$, they call it Bruhat-tits group scheme. I want to know if we have any notion of connection on $G$ torsors on $X$?
 A: Grothendieck has the following interpretation: A connection on a family is the datum that glues fibers over first order neighborhoods.  When you have a $G$-torsor over $X$, a connection will allow you to identify the fibers of the torsor when you move along a tangent vector.
Here is a universal definition: Let $I$ be the ideal sheaf of the diagonal $\Delta: X \hookrightarrow X \times X$, and let $X^{(2)}$ be the subscheme of $X \times X$ cut out by $I^2$.  This is the first order neighborhood of the diagonal.  Composition with the canonical projection maps yields $p_1, p_2: X^{(2)} \to X \times X \to X$.  By pulling back our $G$-torsor $P \to X$, along these maps, we get two $G$-torsors on $X^{(2)}$.  A connection on $\pi$ is an isomorphism $\eta: p_1^* P \to p_2^* P$.  The connection is flat (or integrable) if $p_{13}^* \eta = p_{23}^*\eta \circ p_{12}^*\eta$, where the $p_{ij}$ come from the first order neighborhood of the diagonal in $X^3$, together with projections $X \times X \times X \to X \times X$.
There is also a functorial definition for flat connection using the formalism of crystals.  See, e.g., the notes from Nov. 17 and 19 on this page.
