Suppose that we have a faithful representation $\rm{G}\rightarrow\rm{GL}(V)$ of a semisimple linear algebraic group into a complex vector space $\rm{V}$ of dimension n. Suppose that we have a projective algebraic curve $\rm{X}$ (or just a projective algebraic variety) and a $\rm{G}$-principal bundle over it $\pi:\rm{P}\rightarrow\rm{X}$. We can construct the associated vector bundle $\rm{E}:=(\rm{P}\times\rm{V})/\rm{G}$, where $\rm{G}$ acts on $\rm{P}\times\rm{V}$ as: $$(p,v)\cdot g=(p\cdot g, g^{-1}(v))\text{.}$$ It is supposed to exists a canonical morphism of schemes $$\rm{P}\rightarrow\rm{Isom}(\rm{V}\times\rm{X},\rm{E})$$ but I can´t see what morphism it is. Could you help me? Thank you for your time.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The definition of $\mathrm{Isom}(V\times X, E)$ should be relative to $X$, so that it is a bundle on $X$. Namely, $\mathrm{Isom}(V\times X, E) = \{(x, \phi)\;;\; x\in X, \phi: V\cong E_x\}$. It is then easy to see that $P\to \mathrm{Isom}(V\times X, E)$ should be defined as $P\ni g\mapsto (\pi(g), (v\mapsto (g^{-1}, v))).$
