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

Statement

Setup

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 ${\displaystyle \pi }$.

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 ${\displaystyle G(V)\times H(W)}$.

Global theta correspondence

The global theta lift can be defined on the cuspidal automorphic representations of ${\displaystyle G(V)}$ as well.[1]

Howe duality conjecture

The Howe duality conjecture states that:[2]

(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, ${\displaystyle \pi =\pi '}$.

The Howe duality conjecture for ${\displaystyle E}$ with odd residue characteristic was proved by Jean-Loup Waldspurger in 1990.[3] Wee Teck Gan and Shuichiro Takeda gave a proof in 2014 that works for any residue characteristic.[2]

Etymology

Let ${\displaystyle \theta }$ be the theta correspondence between ${\displaystyle \mathrm {Mp} (2)}$ and ${\displaystyle SO(3)}$. According to Waldspurger (1986) harvtxt error: no target: CITEREFWaldspurger1986 (help), one can associate to ${\displaystyle \theta }$ a function ${\displaystyle f(\theta )}$, which can be proved to be a modular function of half integer weight, that is to say, ${\displaystyle f(\theta )}$ is a theta function.

See also

• Metaplectic group

References

• 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

1. Waldspurger 1991.
2. Gan & Takeda 2014. sfn error: no target: CITEREFGanTakeda2014 (help)
3. Waldspurger 1990.