Howe 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). 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).

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).

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 representation 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). Thanks to their works, we are able to define the local theta correspondence in our situation.

Definition. ( local theta correspondence ). Let Irr(G(V)) ( resp. Irr(H(W)) ) be the set of all irreducible admissible representations of G(V) ( resp. H(W) ). Let θ be the map Irr(G(V)) → Irr(H(W)), which associates every irreducible admissible representation π of G(V) the irreducible admissible representation θ(π,ψ) of H(W). We call θ the local theta correspondence for the pair G(V)×H(W).

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