Friday, 11 December 2009

ag.algebraic geometry - Can there exist two non-equivalent equivariant actions of a group on vector bundle?

Maybe I am miss understanding the question, but it seems the answer is yes.



Take your favorite G-space, mine is S1 with the mathbbZ/2-action "flip". Then consider the trivial vector bundles S1timesV, where V is a G-representation. In my favorite example V=mathbbR can be either the trivial representation or the sign representation. Taking the diagonal G-action gives an equivariant action of the group on the vector bundle. They are distinct for distinct representations (at least in the S1-example) yet the underlying vector bundles are the same if the representations have the same dimension (they are trivial bundles after all).

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