In both cases, it appears possible to adapt the construction to apply to quasisplit unitary groups as well. It is expected similar results could be done for adjoint representations of ${{\mathrm{GL}}}_4$ and ${{\mathrm{GL}}}_5$ by studying these integral representations.

Acknowledgements
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The first named author was supported by NSA grants H98230-15-1-0234 and H98230-16-1-0125. The second named author was supported by NSFC grant 11801577 and a fellowship from the Pacific Institute for the Mathematical Sciences (PIMS). We thank Jack Buttcane, Jim Cogdell, Clifton Cunningham, Sol Friedberg, Paul Garrett, Dorian Goldfeld, Dihua Jiang, Muthukrishnan Krishnamurthy, Xiaoqing Li, Baiying Liu, David Loeffler, Stephen Miller, Michitaka Miyauchi, Ralf Schmidt, Richard Taylor, and Fan Zhou for stimulating questions, helpful suggestions, and useful discussions. We thank Fangyang Tian for sending us his unpublished preprints [@Tianthesis; @Ti18c] and for his comments on our Lemma \[lemnonvanishing\] of a previous draft. We are very grateful to the anonymous referees for their careful reading and many valuable suggestions.

Local zeta integral for the adjoint representation of Lg and Lg
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In this section, we review Ginzburg’s local zeta integral [@G] for the adjoint representation of ${{\mathrm{GL}}}_3$ and prove the local functional equation for these local zeta integrals. Ginzburg’s construction was extended 