Let $G$ be a connected split reductive group over a field $k$ of characteristic $0$. Let $T$ and $T'$ be two split maximal tori of $G$ and $B \supset T, B' \supset T'$ be two Borel subgroups of $G$.

Let $W(G,T)=N_G(T)/T$ and $W(G,T')=N_G(T')/T'$ be the Weyl groups of $G$ relative to $T$ and $T'$ respectively.

Let $w \in W(G,T)$ be the relative position of $B$ and $B'$ i.e. there exists $b \in B$ such that $bB'b^{-1} = wBw^{-1}$ (where I see $w$ as an element of $G$).

My question is : is there an explicit description of the relative position of $B$ and $B'$ as an element of $W(G,T')$ ?

Note that given any element $g$ of $G$ such that $gTg^{-1} = T'$ then the map $\gamma \mapsto g\gamma g^{-1}$ induces a bijection between $W(G,T)$ and $W(G,T')$. Ideally I would like to find a concrete recipe for finding a $g$ such that the relative position of $B$ and $B'$ as an element of $W(G,T')$ is $gwg^{-1}$.


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    $\begingroup$ I don't know what sort of recipe you want; this is something that you can work out case-by-case, but it's probably impossible to say something at this level of generality. $\endgroup$ – skd Apr 20 at 16:41
  • $\begingroup$ I was hoping for answer of the form : take $g \in G$ such that $gTg^{-1} = T'$ and $gBg^{-1} = B'$ then $B$ and $B'$ are in relative position $gwg^{-1}$. But I can't seem to prove this. I still think/hope there should be a general answer. $\endgroup$ – Jdoe Apr 20 at 16:59

I finally found a solution to my own question :

Let $g \in G$ be such that $gTg^{-1} = T'$ and $gwBw^{-1}g^{-1} = B'$. Let $b \in B$ be as in the question i.e. it verifies $bB'b^{-1} = wBw^{-1}$. We obtain $$ gbB'b^{-1}g^{-1} = B' $$ but since $B'$ is a parabolic subgroup it is its own normalizer this implies that $gb \in B'$.

Finaly we find that $$ gbB(gb)^{-1} = gbBb^{-1}g^{-1} = gBg^{-1}= gw^{-1} g^{-1}gwBw^{-1}g^{-1}gwg^{-1} = gwg^{-1}B'gwg^{-1}. $$

We conclude that $B$ and $B'$ are in relative position $gw^{-1}g^{-1}$ with respect to $W(G,T')$ (and $B'$).


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