Weyl groups of Levi factors and their cosets

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 π:PL be the projection, define BL=π(B).

Letting W be the Weyl group of G, define

WL={wW | wBLw1B}.

It is clear that this is well-defined. Letting WL be the Weyl group of L, a paper I am reading claims that WL is a set of coset representatives for W/WL. 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?


Source : Link , Question Author : Sarah , Answer Author : Community

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