## What does it mean to reduce the structure group of a principal bundle?

Let $Fl_G$ be the affine flag variety for a reductive algebraic group $G$. While this is not really a variety, I read that we can describe it as a functor: for a scheme $S$, the set $\operatorname{Hom}(S, Fl_G)$ consists of triples $(F_G, \beta, \epsilon)$, where $F_G$ is a principal $G$-bundle on the affine line and $\beta$ is a trivialization away from the point $0$ (as in the definition of the affine Grassmannian $Gr$), and $\epsilon$ is a reduction of $F_G|_{D_0}$ to $B$, where $B$ is a fixed Borel subgroup of $G$.

I could not understand the meaning of "reduction of $F_G|_{D_0}$ to $B$". What does it mean?

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 I think you should read the FAQ regarding how to pose a question here. In particular there's plenty of notation here that is not defined (the disk for example). Having said so, your F I presume is a principal $G$-bundle on a curve, $B \subset G$ it's Borel subgroup, B-bundles give rise to G-bundles since you can consider $P \rightarrow P \times_B G/B$. Not every $G$-bundle is induced from a $B$ bundle this way. The reduction of the restriction of $F_G$ to the disk $D_0$ is an iso to such a bundle. – Reimundo Heluani Sep 30 2011 at 12:36 thank you very much. S. Carnahan and Reimundo Heluani – yingjin bi Oct 1 2011 at 0:58