I would like to understand an argument in https://link.springer.com/content/pdf/10.1007/BF01393904.pdf, which uses Langlands' result on the constant term of Eisenstein series, but I'm not getting it and I would appreciate any explanation.

On page 429 they define $$ c(w, \lambda) = \int_{ (w' N_0(\mathbb{A})(w')^{-1} \cap N_0 (\mathbb{A})) \backslash N_0(\mathbb{A}) } exp(<H_{P_0} ((w')^{-1}n) , \lambda+ \rho_{P_0} >) dn. $$ Then they say $$ \int_{N_0(\mathbb{Q}) \backslash N_0(\mathbb{A})} E^G_{P_0}(\lambda, ng) dn = \sum_{w \in W} c(w, \lambda) exp( <H_{P_0} (g) , w \lambda + \rho_{P_0} > ) dn. $$ I think that this statement should follow from the Langlangds' result on page 11 in these slides by Shahidi with $P = P' = P_0$:
https://www.ima.umn.edu/materials/2018-2019/SW11.14-16.18/27644/Shahidi_Abhi.pdf (or another reference for this is Theorem 6.2.2. (p114) in Shahidi's book "Eisenstein series and autompic L-functions"). But I seem to be missing something and have not been able to deduce the above statement. For example, in the above $c(w, \lambda)$ is independent of $g$ while in the formula in the link it seems to depend on it... Any explanation would be appreciated. Thank you.

Some notation: $P_0$ is 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 $$ E^G_{P_0}(\lambda, g) = \sum_{\gamma \in P(\mathbb{Q}) \backslash G(\mathbb{Q}) } exp(<\lambda + \rho_{P_0}, H_{P_0} (\gamma g) >). $$

PS I am starting to wonder maybe $c(w,\lambda)$ in the paper (as above) is actually a typo(?) and that it should be replaced by $$ M(w, \lambda)(1)(g) = \int_{ (w' N_0(\mathbb{A})(w')^{-1} \cap N_0 (\mathbb{A})) \backslash N_0(\mathbb{A}) } exp(<H_{P_0} ((w')^{-1}ng) , \lambda+ \rho_{P_0} >) exp(<H_{P_0} (g) , -( w \lambda+ \rho_{P_0}) >) dn $$ as in the formula given in the link above. I am new to the subject and I'm not too sure as maybe I'm missing stuff. Any comments would be appreciated. Thank you.


I haven't carefully gone through your notations, but usually the constant term of Eisenstein series can be expressed as a sum, going over $w$ in some subset of Weyl group, of values of sections after applying appropriate intertwining operators. The last formula of $M(w,\lambda)$ seems about right to me.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.