On equivariant Serre problem for principal bundles

Let $E_G$ be a $\mathrm{\Gamma }$–equivariant algebraic principal $G$–bundle over a normal complex affine variety $X$ equipped with an action of $\mathrm{\Gamma }$, where $G$ and $\mathrm{\Gamma }$ are complex linear algebraic groups. Suppose $X$ is contractible as a topological $\mathrm{\Gamma }$–space with a dense orbit, and $x_0 \in X$isa$\Gamma$--fixedpoint\mathrm{.}Weshowthatif$\Gamma$isreductive,then$E_G$admitsa$\Gamma$--equivariantisomorphismwiththeproductprincipal$G$--bundle$X \times_{\rho} E_G(x_0)$,where$\rho,:, \Gamma , \longrightarrow, G$ is a homomorphism between algebraic groups. As a consequence, any torus equivariant principal $G$-bundle over an affine toric variety is equivariantly trivial. This leads to a classification of torus equivariant principal $G$-bundles over any complex toric variety.