I know Saito-Kurokawa(SK) representation is the famous non-tempered representation of $SO(5)$. But since the tempered or non-tempered terms are concerned with local phenomenon, I am wondering that when people says SK is non-tempered, are they meaning that it is non-tempered at every places or non-tempered for all but finite places?


I want to ask one more simple quetion. Does every supercuspidal representation should be automatically tempered? I think it should be by definition.



On the Saito Kurokawa representation
Under the isomorphism $\mathrm{PGSp}(4) \cong \mathrm{SO}(5)$ the Saito-Kurokawa representation corresponds to a Siegel modular form. In particular its archimedean component is in the holomorphic discrete series, so it is tempered. All non-archimedean components are non-tempered.


Also, yes: supercuspidal implies discrete series implies tempered.


