Tannaka duality on quotient stacks

Let = [X/G] be the quotient stack of a scheme Xby an affine group scheme Gover a field k. Assume that there is a line bundle on whose underlying line bundle on Xis very ample. Let VB() be the category of vector bundles on .We show that is canonically isomorphic to the stack of fiber functors on VB(). This is an analogue of the Tannaka duality for affine groups.Let = [X/G] be the quotient stack of a scheme Xby an affine group scheme Gover a field k. Assume that there is a line bundle on whose underlying line bundle on Xis very ample. Let VB() be the category of vector bundles on .We show that is canonically isomorphic to the stack of fiber functors on VB(). This is an analogue of the Tannaka duality for affine groups.