# Relative position and change of torus

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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• 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. – skd Apr 20 at 16:41
• 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. – Jdoe Apr 20 at 16:59

## 1 Answer

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 $$B'$$).