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). Theta correspondence is a generalisation of Shimura correspondence. It's also named as Howe conjecture. Theta correspondence is a correspondence between certain representations on Mp(2n) and these on SO(2n+1). In fact, it's not yet fully constructed. The case n = 1 is already completely proved by J. L. Waldspurger in Waldspurger (1980) and Waldspurger (1991).
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Local Theta Correspondence
Given a dual pair (G,H) defined over a local field F, there is a relation between the irreducible admissible representations Irr(G) and Irr(H).
Howe Conjecture
Let E be a nonarchimedean local field of characteristic not 2, with its quotient field of charcateristic p. Let F be a quadratic extension over E. Let V ( resp. W ) be a n-dimensional hermitian space ( resp. a m-dimensional hermitian space) over F. We assume further G(V) ( resp. H(W) ) to be the isometry group of V ( resp. W ). There exists a Weil respresentation associated to a non-trivial additive character ψ of F for the pair G(V)×H(W), which we write as ρ(ψ). Let π be a irreducible admissible representation of G(V). Here, we only consider the case G(V)×H(W) = SO(n)×SO(m) or U(n)×U(m). We can find a certain representation θ(π,ψ) of H(W), which is in fact a certain quotient of the Weil representation ρ(ψ) by π. Now we are ready to state the conjecture of Howe.
Howe Conjecture.
(i) θ(π,ψ) is irreducible or 0;
(ii) Let π, π' be two irreducible admissible representations of G(V), s. t. θ(π,ψ) = θ(π',ψ) ≠ 0. Then, π = π'.
Here, we have used the notations of Gan & Takeda (2014). The Howe conjecture in our setting was already prove by J. L. Waldspurger in Waldspurger (1990). W. T. Gan and S. Takeda reprove it, using a simpler and more uniform method, in Gan & Takeda (2014).
Reason for the name
Let θ be the theta correspondence between Mp(2) and SO(3). According to Waldspurger (1986), one can associate to θ a function f(θ), which can be proved to be a modular function of half integer weight, that is to say, f(θ) is a theta function.