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Let $P_0$ be a minimal $\mathbb{Q}$-parabolic subgroup of $G$, a semisimple linear algebraic group over $\mathbb{Q}$. Then $P_0 = M_0 N_0$ where $M_0$ is a Levi subgroup of $P_0$. Let $E^G_{P_0}$ be the Eisenstein series $$ E^G_{P_0}(\lambda, g) = \sum_{\gamma \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) } \exp(\langle\lambda + \rho_{P_0}, H_{P_0} (\gamma g)\rangle). $$ Let $$ c(w, \lambda) = \int_{ (w' N_0(\mathbb{A})(w')^{-1} \cap N_0 (\mathbb{A})) \backslash N_0(\mathbb{A}) } \exp(\langle H_{P_0} ((w')^{-1}n) , \lambda+ \rho_{P_0}\rangle) dn. $$ I am wondering what is meant by "the singular hyperplanes of $c(w, \cdot)$"?

This is in Lemma 7 (p 429) of the paper Franke, Manin, and Tshinkel - Rational points of bounded height on Fano varieties, which states: The singular hyperplanes of $c(w, \cdot)$ containing $\rho_{P_0}$ are precisely the hyperplanes $$ \langle\check{\alpha}, \lambda - \rho_{P_0}\rangle = 0 $$ where $\alpha$ is a simple positive root such that $w \alpha$ is a negative root.

I would like to understand the statement of this lemma. I have been going through definitions but I have not been able to do this yet. I would greatly appreciate any explanation on this.

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    $\begingroup$ I think that it just indicates the hyperplanes containing a point $\lambda$ at which the meromorphic function $\lambda \mapsto c(w, \lambda)$ has a pole. $\endgroup$
    – LSpice
    Commented May 30, 2019 at 14:00
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    $\begingroup$ But I would like to thank the authors for their reference just to [4], §7, which is nearly a third of the book (75 pages). $\endgroup$
    – LSpice
    Commented May 30, 2019 at 14:02
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    $\begingroup$ My hypothesised meaning is consistent with the reference on pp. 170–171 of Langlands's book, which supposes that "$\phi(\cdot)$ is a function meromorphic on [bizarre handwritten symbol] whose singularities lie along hyperplanes of the form $\alpha(H) = \mu$ where $\alpha$ is a real linear function [on] $\mathfrak a$ and $\mu$ is a complex number", and then immediately refers in the next sentence to the singular hyperplanes of $\phi(\cdot)$. $\endgroup$
    – LSpice
    Commented May 30, 2019 at 14:07
  • $\begingroup$ @LSpice I must have a different version of the book as I can not find it on pp 170-171 of the book. but thank you for your help on this. It is appreciated! $\endgroup$
    – Johnny T.
    Commented May 30, 2019 at 14:14

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