# 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 and those on the special orthogonal group . The case was constructed by Jean-Loup Waldspurger in Waldspurger (1980) and Waldspurger (1991).

## Statement[edit]

### Setup[edit]

Let be a non-archimedean local field of characteristic not , with its quotient field of characteristic . Let be a quadratic extension over . Let (respectively ) be an -dimensional Hermitian space (respectively an -dimensional Hermitian space) over . We assume further (resp. ) to be the isometry group of (resp. ). There exists a Weil representation associated to a non-trivial additive character of for the pair , which we write as . Let be an irreducible admissible representation of . Here, we only consider the case or . We can find a certain representation of , which is in fact a certain quotient of the Weil representation by .

### Howe duality conjecture.[edit]

(i) is irreducible or ;

(ii) Let be two irreducible admissible representations of , such that . Then, .

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 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 (respectively ) be the set of all irreducible admissible representations of (respectively ). Let be the map , which associates every irreducible admissible representation of the irreducible admissible representation of . We call the local theta correspondence for the pair .

**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 as well, see Waldspurger (1991).

## Etymology[edit]

Let be the theta correspondence between and . According to Waldspurger (1986), one can associate to a function , which can be proved to be a modular function of half integer weight, that is to say, is a theta function.

## See also[edit]

## References[edit]

- Howe, Roger (1979), "θ-series and invariant theory" (PDF), in Borel, Armand; Casselman, W. (eds.),
*Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1*, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 275–285, ISBN 978-0-8218-1435-2, MR 0546602 - Mœglin, Colette; Vignéras, Marie-France; Waldspurger, Jean-Loup (1987),
*Correspondances de Howe sur un corps p-adique*, Lecture Notes in Mathematics,**1291**, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0082712, ISBN 978-3-540-18699-1, MR 1041060 - Waldspurger, Jean-Loup (1987), "Représentation métaplectique et conjectures de Howe",
*Astérisque*, Séminaire Bourbaki 674, 152-153: 85–99, ISSN 0303-1179, MR 0936850 - Waldspurger, Jean-Loup (1980), "Correspondance de Shimura",
*J. Math. Pures Appl.*,**59**(9): 1–132 - Waldspurger, Jean-Loup (1991), "Correspondances de Shimura et quaternions",
*Forum Math.*,**3**(3): 219–307, doi:10.1515/form.1991.3.219 - Waldspurger, Jean-Loup (1990), "Démonstration d'une conjecture de dualité de Howe dans le cas p-adique, p ≠ 2",
*Festschrift in Honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I*, Israel Math. Conf. Proc.,**2**: 267–324 - Gan, Wee Teck; Takeda, Shuichiro (2016), "A proof of the Howe duality conjecture" (PDF),
*J. Amer. Math. Soc.*,**29**(2): 473-493. - Gan, Wee Teck; Li, Wen-Wei,
*The Shimura-Waldspurger correspondence for Mp(2n)*, arXiv:1612.05008, Bibcode:2016arXiv161205008T