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@DaveAnderson Thank you very much! By the way where can I find the relation between the $K_0$ of $X$ and the normalization of $X$? Is it also in Weibel's book?
Yes I agree. I think that it is a little bit more complicated to express the quasi-representability under the self-duality but I'm not sure about the details.
@user74230 Thank you very much for your enlightening comment! Is the result true if we consider the cohomology on compact Stein manifold instead of arbitrary Stein manifold?
Actually $\varphi$ gives the "$G$-action" on $\mathcal{L}$ and this is what does it mean by $G$-equivariant vector bundles or more generally, $G$-equivariant sheaves: We such a map $\varphi$ which satisfies some properties. See for example Kashiwara's paper kurims.kyoto-u.ac.jp/~kenkyubu/kashiwara/sd.pdf page 22-23.
$\sigma^* s$ is defined simply by "composition with $\sigma$". We can see that if $s$ is a section of $\mathcal{L}$ then the composition gives a section of $\sigma^* \mathcal{L}$.
