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Let $G$ be a semisimple algebraic group defined over an algebraic closed field. Fix a Borel subgroup $B$ and a maximal torus $T\subset B$. Let $\Delta$ be the set of simple roots. For a subset $\Delta_\gamma\subset\Delta$, denote by $P_\gamma$ the standard parabolic associated with $\Delta_\gamma$ and $P^{-}_\gamma$ be the opposite one.

Suppose now $\Delta_\gamma\cup\Delta_\mu=\Delta$, can we get $P_\gamma^{-}\cdot P_{\mu}=G$?

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  • $\begingroup$ My first instinct was no, but this is kind of a pain to check, even for the non-trivial $\operatorname{GL}_3$ case! \\ Equivalent questions, the first suggested by a colleague to whom I mentioned the question: is $P_\gamma^-\backslash G/P_\mu$ a singleton? Is your product closed? Is it a group? The first might be easier to check if we conjugate $P_\gamma^-$ to a standard parabolic; the last could conceivably be checked by grungy computations with Bruhat cells. $\endgroup$
    – LSpice
    Commented Aug 12 at 22:36
  • $\begingroup$ @LSpice Thanks! $\endgroup$
    – fool rabbit
    Commented Aug 13 at 5:39

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