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Let $G$ be a connected semisimple algebraic group defined over an algebraically closed field $F$ of characteristic $0$. Let $B$ be a minimal parabolic subgroup of $G$ (i.e. a Borel subgroup), let $P$ be a parabolic subgroup of $G$ containing $B$, and let $L$ be a Levi factor of $P$. Letting $\pi\colon P \rightarrow L$ be the projection, define $B_L = \pi(B)$.

Letting $W$ be the Weyl group of $G$, define

$$W^L = \{\text{$w \in W$ $|$ $w B_L w^{-1} \subset B$}\}.$$

It is clear that this is well-defined. Letting $W_L$ be the Weyl group of $L$, a paper I am reading claims that $W^L$ is a set of coset representatives for $W/W_L$. I can verify this for many examples (for instance, general linear and symplectic groups), but I don't see how to do it in general. Can someone either explain this or give a reference?

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  • $\begingroup$ By definition, $B_L$ is a $quotient$ of $B$, and therefore, it needs some clarification to say $wB_Lw^{-1} \subset B$. One may have to choose a $T$ -equivariant splitting of the map $P\rightarrow L$, and identify $L$ with the image of the splitting. $\endgroup$
    – Venkataramana
    Mar 5, 2017 at 12:53
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    $\begingroup$ Once you do that, fix any element $w\in W$. The group $wBw^{-1}$ (which corresponds to $some$ positive system of roots), contains either a positive root of $B_L$ or a negative root of $B_L$. Hence its intersection with $L$ (image of splitting) will be a Borel $B'_L$ subgroup of $L$ containing $T$. Thus $B'_L$ is conjugate to $B_L$ by a Weyl group element of $L$. After changing $w$ by a Weyl group element of $L$, we may assume that $w^{-1}Bw$ contains $B_L$. This is what you wanted. $\endgroup$
    – Venkataramana
    Mar 5, 2017 at 12:58

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