Theta correspondence

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 ${\displaystyle \pi }$.

(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) harvtxt error: no target: CITEREFGanTakeda2014 (help). 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) harvtxt error: no target: CITEREFGanTakeda2014 (help) 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).