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Oct 16, 2016 at 14:44 comment added Tyler Lawson I'm not 100% clear on the question. Can you please clarify which diagram of parabolics that you mean? The group $G$ is maximal among parabolic subgroups and so, if you take the hocolim over all parabolics, you just recover the value on the maximal object $BG$. If you are restricting attention to proper parabolics then this is a different question. However, you mention having a Borel pair and so it seems like you might want to restrict to either parabolics containing $B$ or those containing $T$; in the former case it seems like a counterexample is given by $GL_2$.
S Oct 16, 2016 at 6:24 history suggested AHusain CC BY-SA 3.0
Typos and classifying-spaces tag
Oct 16, 2016 at 6:02 review Suggested edits
S Oct 16, 2016 at 6:24
Oct 16, 2016 at 5:56 history edited prochet CC BY-SA 3.0
added 38 characters in body
Oct 16, 2016 at 4:24 history asked prochet CC BY-SA 3.0