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.

Let ${\displaystyle F}$ be a local or a global field, not of characteristic ${\displaystyle 2}$. Let ${\displaystyle W}$ be a symplectic vector space over ${\displaystyle F}$, and ${\displaystyle Sp(W)}$ the symplectic group.

Fix a reductive dual pair ${\displaystyle (G,H)}$ in ${\displaystyle Sp(W)}$. There is a classification of reductive dual pairs.

${\displaystyle F}$ is now a local field. Fix a non-trivial additive character ${\displaystyle \psi }$ of ${\displaystyle F}$. There exists a Weil representation of the metaplectic group ${\displaystyle Mp(W)}$ associated to ${\displaystyle \psi }$, which we write as ${\displaystyle \omega _{\psi }}$.

Given the reductive dual pair ${\displaystyle (G,H)}$ in ${\displaystyle Sp(W)}$, one obtains a pair of commuting subgroups ${\displaystyle ({\widetilde {G}},{\widetilde {H}})}$ in ${\displaystyle Mp(W)}$ by pulling back the projection map from ${\displaystyle Mp(W)}$ to ${\displaystyle Sp(W)}$.

The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of ${\displaystyle {\widetilde {G}}}$ and certain irreducible admissible representations of ${\displaystyle {\widetilde {H}}}$, obtained by restricting the Weil representation ${\displaystyle \omega _{\psi }}$ of ${\displaystyle Mp(W)}$ to the subgroup ${\displaystyle {\widetilde {G}}\cdot {\widetilde {H}}}$. The correspondence was defined by Roger Howe in Howe (1979). The assertion that this is a 1-1 correspondence is called the Howe duality conjecture.

Key properties of local theta correspondence include its compatibility with Bernstein-Zelevinsky induction and conservation relations concerning the first occurrence indices along Witt towers .