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Remark 3.3 in Behrend and Dhillon's paper "On the motivic class of the stack of bundles" says $\mu(BP) \mu(G) =\mu(G/P) $ for all parabolic subgroups of G, where this requires the torsor relations only for the group $G$. I wonder how is this proved, do we show that the natural map $G/P\rightarrow */P$ is a $G$-bundle? If yes, could you give me a few hints on how to show it's a $G$-bundle. Thank you for your time.

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  • $\begingroup$ This map is a quotient of the trivial $G$-bundle $G\to *$, no? (under some functor from $P$-{stacks or whatever} to {stacks or whatever}) Depending on your exact setup I suppose there might be descent conditions to check, but I would expect that those should be straightforward over a point. $\endgroup$ Feb 15, 2021 at 4:32
  • $\begingroup$ @Tabes Bridges Thanks, let me work on your suggestion. Thanks! $\endgroup$
    – user567863
    Feb 15, 2021 at 20:30

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