# Theta correspondence

In mathematics, the theta correspondence or Howe correspondence is a correspondence between automorphic forms associated to the two groups of a dual reductive pair, introduced by Howe (1979) as a generalisation of the Shimura correspondence. It is a conjectural correspondence between certain representations on the metaplectic group ${\displaystyle \mathrm {Mp} (2n)}$ and those on the special orthogonal group ${\displaystyle \mathrm {SO} (2n+1)}$. The case ${\displaystyle n=1}$ was constructed by Jean-Loup Waldspurger in Waldspurger (1980) and Waldspurger (1991).

## Statement

### Setup

Let ${\displaystyle E}$ be a non-archimedean local field of characteristic not ${\displaystyle 2}$, with its quotient field of characteristic ${\displaystyle p}$. Let ${\displaystyle F}$ be a quadratic extension over ${\displaystyle E}$. Let ${\displaystyle V}$ (respectively ${\displaystyle W}$) be an ${\displaystyle n}$-dimensional Hermitian space (respectively an ${\displaystyle m}$-dimensional Hermitian space) over ${\displaystyle F}$. We assume further ${\displaystyle G(V)}$ (resp. ${\displaystyle H(W)}$) to be the isometry group of ${\displaystyle V}$ (resp. ${\displaystyle W}$). There exists a Weil representation associated to a non-trivial additive character ${\displaystyle \psi }$ of ${\displaystyle F}$ for the pair ${\displaystyle G(V)\times H(W)}$, which we write as ${\displaystyle \rho (\psi )}$. Let ${\displaystyle \pi }$ be an irreducible admissible representation of ${\displaystyle G(V)}$. Here, we only consider the case ${\displaystyle G(V)\times H(W)=\mathrm {SO} (n)\times \mathrm {SO} (m)}$ or ${\displaystyle U(n)\times U(m)}$. We can find a certain representation ${\displaystyle \theta (\pi ,\psi )}$ of ${\displaystyle H(W)}$, which is in fact a certain quotient of the Weil representation ${\displaystyle \rho (\psi )}$ by ${\displaystyle \pi }$.

### Howe duality conjecture.

(i) ${\displaystyle \theta (\pi ,\psi )}$ is irreducible or ${\displaystyle 0}$;

(ii) Let ${\displaystyle \pi ,\pi '}$ be two irreducible admissible representations of ${\displaystyle G(V)}$, such that ${\displaystyle \theta (\pi ,\psi )=\theta (\pi ',\psi )\neq 0}$. Then, ${\displaystyle \pi =\pi '}$.

Here, we have used the notations of Gan & Takeda (2014); the Howe conjecture in this setting was proved by J. L. Waldspurger in Waldspurger (1990) when the residue characteristic of ${\displaystyle E}$ is odd, and later W. T. Gan and S. Takeda gave a proof in Gan & Takeda (2014) which works for any residue characteristic.

Definition. (local theta correspondence). Let ${\displaystyle \mathrm {Irr} (G(V))}$ (respectively ${\displaystyle \mathrm {Irr} (H(W))}$) be the set of all irreducible admissible representations of ${\displaystyle G(V)}$ (respectively ${\displaystyle H(W)}$). Let ${\displaystyle \theta }$ be the map ${\displaystyle \mathrm {Irr} (G(V))\rightarrow \mathrm {Irr} (H(W))}$, which associates every irreducible admissible representation ${\displaystyle \pi }$ of ${\displaystyle G(V)}$ the irreducible admissible representation ${\displaystyle \theta (\pi ,\psi )}$ of ${\displaystyle H(W)}$. We call ${\displaystyle \theta }$ the local theta correspondence for the pair ${\displaystyle G(V)\times H(W)}$.

Comment. Here we can only define the theta correspondence locally, basically because the Weil representation used in our construction is only defined locally; the global theta lift can be defined on the cuspidal automorphic representations of ${\displaystyle G(V)}$ as well, see Waldspurger (1991).

## Etymology

Let ${\displaystyle \theta }$ be the theta correspondence between ${\displaystyle \mathrm {Mp} (2)}$ and ${\displaystyle SO(3)}$. According to Waldspurger (1986), one can associate to ${\displaystyle \theta }$ a function ${\displaystyle f(\theta )}$, which can be proved to be a modular function of half integer weight, that is to say, ${\displaystyle f(\theta )}$ is a theta function.