My question is the title.
In some literature, authors seem to use this without assumption.
Is it ture in general?
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1$\begingroup$ It would be helpful if you could please cite some sources where you have seen this used. $\endgroup$– Uriya FirstCommented Mar 13, 2020 at 12:58
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1$\begingroup$ @Uriya, I saw this in the hypotheses of the covering morphism $pr$ in I.1.3 of the book 'Spectral decomposition and Eisenstein series' by Moeglin and Waldspurger. I guess that the authors made this hypotheses keeping the double covering map from metaplectic group to symplectic group in their mind. $\endgroup$– MontyCommented Mar 13, 2020 at 13:22
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$\begingroup$ Yes. This is true in general. In fact, the definition of automorphic forms on Mp depends on this splitting. If my memory is correct, Moeglin and Waldspurger dealt with more general case than the metaplectic double cover Mp. $\endgroup$– Q-ZhCommented Mar 23, 2020 at 2:35
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$\begingroup$ @Qing, Thank you very much! I referee the book ‘Spectral decomposition and Eisenstein series' by Moeglin and Waldspurger but couldn’t find proof of the generalization of this. Would you please let me know the page of the book? $\endgroup$– MontyCommented Mar 23, 2020 at 15:54
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$\begingroup$ I don't think Moegline-Waldspurger has a proof. The proof is probably in one of the two Weil's classical papers. The point is, MW has to assume there is such a splitting because they considered more general covers. For those more general covers, the existence of such splittings is not guaranteed. Since you only ask the metaplectic cover, the answer should be yes. $\endgroup$– Q-ZhCommented Mar 23, 2020 at 16:18
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