Theta correspondence

In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorphic representations over a global field.

The theta correspondence was introduced by Roger Howe in Howe (1979). Its name arose due to its origin in André Weil's representation theoretical formulation of the theory of theta series in Weil (1964). The Shimura correspondence as constructed by Jean-Loup Waldspurger in Waldspurger (1980) and Waldspurger (1991) may be viewed as an instance of the theta correspondence.