Theta correspondence

Summary

The theta correspondence, or Howe correspondence, matches irreducible representations of the two members of a reductive dual pair inside a symplectic group. The match is read off from the restriction of the oscillator representation of the metaplectic group to the product of the two groups. Roger Howe introduced reductive dual pairs and conjectured that this restriction gives a bijection, now called Howe duality. Globally, the correspondence is realised by theta lifts of automorphic forms. These lifts are a classical way to build automorphic forms on one group from those on another.