On equivariant Serre problem for principal bundles

Let EG be a-equivariant algebraic principal G-bundle over a normal complex affine variety X equipped with an action of , where G and are complex linear algebraic groups. Suppose X is contractible as a topological -space with a dense orbit, and x0 X is a -fixed point. We show that if is reductive, then EG admits a -equivariant isomorphism with the product principal G-bundle X × EG(x0), where : 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, generalizing the main result of [A classification of equivariant principal bundles over nonsingular toric varieties, Internat. J. Math. 27(14) (2016)].