Assume X is a real smooth manifold with an action of the real Lie group G. Let E be a G-vector bundle over X. Consider the spaces of generalized sections over X of E, and of E^* (fiberwise dual). My question is: do the spaces of G-invariants of those two spaces have equal dimension? Of course one should assume something about the nature of the action of G on X, in particular so that both dimensions are finite. If G acts transitively, an invariant secion is automatically smooth, and this is a correct and easy statement . What if G acts with finitely many orbits which are geometrically reasonable? What if we are in an algebraic variety-algebraic group setting?
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$\begingroup$ I am not sure what you mean in your last sentence, but in the setting of algebraic geometry this is definitely false : a very ample line bundle has a lot of sections while its dual has none. $\endgroup$– abxJan 26, 2015 at 7:01
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$\begingroup$ Could you please give an example over the reals? $\endgroup$– DimaJan 26, 2015 at 9:04
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$\begingroup$ Take the projective space and the line bundle $\mathcal{O}(1)$. $\endgroup$– abxJan 26, 2015 at 9:36
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$\begingroup$ Sorry, I don't understand. What is the group $G$ in this example? $\endgroup$– DimaJan 26, 2015 at 14:57
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$\begingroup$ The trivial group. $\endgroup$– abxJan 26, 2015 at 15:16
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