Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a Borel pair $(B,T)$. Let $BG$ be the classifying space of $G$.
Can we say that $BG$ is the homotopy colimit of all $BP$ for $P$ a standard parabolic?
$\begingroup$
$\endgroup$
1
-
1$\begingroup$ 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 $GL_2$. $\endgroup$– Tyler LawsonCommented Oct 16, 2016 at 14:44
Add a comment
|