The Saito-Kurokawa lifting and functoriality

Certain nontempered liftings from PGL (2) × PGL (2) to PGSp (4) are constructed using the theory of (local and global) theta lifts. The resulting representations on PGSp (4) are the Saito-Kurokawa representations. The lifting is shown to be functorial under certain reasonable assumptions on the local Langlands correspondence for PGSp (4).