# Coset representatives for $wPw^{-1} \cap P' \backslash P'$ for parabolic subgroups $P'$ and $P$?

I'm going through the calculation of the constant term of Eisenstein series in Moeglin and Waldsburger's Spectral Decomposition and Eisenstein series book (section II.1.7), and am confused on a small detail. Let $$P = MN$$ and $$P' = M'N'$$ be standard parabolic subgroups of a connected, reductive group $$G$$ over a field $$k$$, identified with their $$k$$-points. If $$w$$ is an element of the Weyl group, for which $$w.\alpha > 0$$ for all simple roots of $$M$$, and $$w^{-1}.\alpha > 0$$ for all simple roots of $$M'$$, I'm trying to understand how to decompose

$$P' \cap wPw^{-1} \backslash P'$$ The calculation of the constant term appears to rely on the fact that a set of right coset representatives of $$P' \cap wPw^{-1}$$ in $$P'$$ is $$m'n'$$, where $$m'$$ runs through a set of right coset representatives of $$P' \cap wMw^{-1} \backslash M'$$, and $$n'$$ runs through a set of right coset representatives of $$P' \cap m'^{-1}wNw^{-1}m' \backslash N'$$. I have been trying for awhile to prove that this is true, but have gotten stuck. I would appreciate any hint, proof, or reference.

• Please try to avoid creating new tags without first seeking community input. math.meta.stackexchange.com/q/34525 Jul 1 at 1:14
• @XanderHenderson, was there a tag that you deleted? The three that I see now, "algebraic groups", "reductive groups", and "langlands program" seem reasonable. For that matter, further tags about Bruhat decompositions and similar would surely be reasonable, since these are admittedly nasty-technical things that do have a big impact. Can you clarify? Jul 1 at 1:26
• @paulgarrett it looks like the question was originally tagged with eisenstein-series, which had no other questions under its heading Jul 1 at 1:32
• @AtticusStonestrom, hm... well, this question is very relevant to Eisenstein series stuff... there should be other things here under "Eisenstein series", I'd think, but, ... well, it's too late in the day to tilt at any windmills... :) Still, there have been many questions here that would deserve the "E-series" label... maybe some day someone should go through and attend to that. Not today, for me. :) Jul 1 at 1:41
• @paulgarrett I don't disagree, but the tag did not exist prior to this question, and it is general practice to discuss new tags before introducing them. Jul 2 at 22:16