On the theta correspondence for (𝐺𝑆𝑝(4),𝐺𝑆𝑂(4,2)) and Shalika periods

Abstract

We consider both local and global theta correspondences for G S p 4 \mathrm {GSp}_4 and G S O 4 , 2 \mathrm {GSO}_{4,2} . Because of the accidental isomorphism P G S O 4 , 2 ≃ P G U 2 , 2 \mathrm {PGSO}_{4,2} \simeq \mathrm {PGU}_{2,2} , these correspondences give rise to those between G S p 4 \mathrm {GSp}_4 and G U 2 , 2 \mathrm {GU}_{2,2} for representations with trivial central characters. In the global case, using this relation, we characterize representations with trivial central character, which have Shalika period on G U ( 2 , 2 ) \mathrm {GU}(2,2) by theta correspondences. Moreover, in the local case, we consider a similar relationship for irreducible admissible representations without an assumption on the central character.