Skip to main content
MathOverflow

Return to Question

<> -> \langle\rangle, deleted 'Thanks', and other small changes
Source Link
LSpice
LSpice
  • 12.9k
  • 4
  • 45
  • 69

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

This is in Lemma 7 (p 429) of the paper https://link.springer.com/content/pdf/10.1007/BF01393904.pdfFranke, 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 $$ <\check{\alpha}, \lambda - \rho_{P_0}> = 0 $$$$ \langle\check{\alpha}, \lambda - \rho_{P_0}\rangle = 0 $$ 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. Thank you very much.

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

This is in Lemma 7 (p 429) of the paper https://link.springer.com/content/pdf/10.1007/BF01393904.pdf which states: The singular hyperplanes of $c(w, \cdot)$ containing $\rho_{P_0}$ are precisely the hyperplanes $$ <\check{\alpha}, \lambda - \rho_{P_0}> = 0 $$ 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. Thank you very much.

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.

Source Link
Johnny T.
Johnny T.
  • 3.6k
  • 14
  • 29

What is meant by singular hyperplane of $c(w, \cdot)$? (global intertwining operator related to Eisenstein series)

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

This is in Lemma 7 (p 429) of the paper https://link.springer.com/content/pdf/10.1007/BF01393904.pdf which states: The singular hyperplanes of $c(w, \cdot)$ containing $\rho_{P_0}$ are precisely the hyperplanes $$ <\check{\alpha}, \lambda - \rho_{P_0}> = 0 $$ 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. Thank you very much.