# Natural morphism to the scheme of isomorphism

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.

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 $$P\ni g\mapsto (\pi(g), (v\mapsto (g^{-1}, v))).$$