---
tag: ZBGT50BW0Z
kind: concept
name: Theta correspondence
msc: [11F27, "22E46", "22E50"]
sources:
  - url: https://en.wikipedia.org/wiki/Theta_correspondence
same_as:
  wikidata: Q7783546
  wikipedia: 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|CCR7BAQ5N5]] 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.

## Explanations

### L1 human

Take two groups that sit inside a large symplectic group and commute with each other, each being the full commutant of the other.
The oscillator representation, restricted to the pair, breaks into pieces of the form "a representation of the first group times a representation of the second".
The theorem behind the correspondence says that in each piece the first representation determines the second, and conversely.
So the oscillator representation acts as a dictionary between the representations of the two groups.