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 $\mathrm {Mp} (2n)$ and those on the special orthogonal group $\mathrm {SO} (2n+1)$. The case $n=1$ was constructed by Jean-Loup Waldspurger in Waldspurger (1980) and Waldspurger (1991).

Statement

Setup

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

Local theta correspondence

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

Global theta correspondence

The global theta lift can be defined on the cuspidal automorphic representations of $G(V)$ as well.

Howe duality conjecture

The Howe duality conjecture states that:

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

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

The Howe duality conjecture for $E$ with odd residue characteristic was proved by Jean-Loup Waldspurger in 1990. Wee Teck Gan and Shuichiro Takeda gave a proof in 2014 that works for any residue characteristic.

Etymology

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

See also

• Metaplectic group

References

• Howe, Roger (1979), "θ-series and invariant theory", 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 {{citation}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)
• Mœglin, Colette; Vignéras, Marie-France; Waldspurger, Jean-Loup (1987), Correspondances de Howe sur un corps p-adique, Lecture Notes in Mathematics, vol. 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