Theta correspondence

In mathematics, the theta correspondence or Howe correspondence is a correspondence between irreducible admissible representations over a local field and a correspondence of irreducible automorphic representations over a global field, associated to two groups of a reductive dual pair, introduced by Roger Howe in Howe (1979). The name arose due to its origin in André Weil’s representation theoretical formulation of the theory of theta series in Weil (1964). The Shimura correspondence as constructed by Jean-Loup Waldspurger in Waldspurger (1980) and Waldspurger (1991) may be viewed as an instance of theta correspondence.

Statement

Setup

Let $F$ be a local field of characteristic zero. Fix a non-trivial additive character $\psi$ of $F$. Let $W$ be a symplectic vector space over $F$, and $Sp(W)$ the symplectic group.

Given a reductive dual pair $(G,H)$ in $Sp(W)$, one obtains a reductive dual pair $({\widetilde {G}},{\widetilde {H}})$ in the metaplectic group $Mp(W)$ by pulling back the projection map from $Mp(W)$ to $Sp(W)$.

Local theta correspondence

There exists a Weil representation of the metaplectic group $Mp(W)$ associated to $\psi$, which we write as $\omega _{\psi }$.

The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of ${\widetilde {G}}$ and certain irreducible admissible representations of ${\widetilde {H}}$, obtained by restricting the Weil representation $\omega _{\psi }$ of $Mp(W)$ to the subgroup ${\widetilde {G}}\cdot {\widetilde {H}}$. The correspondence was defined by Roger Howe in Howe (1979). The assertion that this is a 1-1 correspondence is called the Howe duality conjecture.

Global theta correspondence

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

Howe duality conjecture

Define ${\mathcal {R}}({\widetilde {G}})$ the set of irreducible admissible representations of ${\widetilde {G}}$, which can be realized as quotients of $\omega _{\psi }$. Define ${\mathcal {R}}({\widetilde {H}})$ and ${\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}})$, likewise.

The Howe duality conjecture asserts that ${\mathcal {R}}({\widetilde {G}}\cdot {\widetilde {H}})$ is the graph of a bijection between ${\mathcal {R}}({\widetilde {G}})$ and ${\mathcal {R}}({\widetilde {H}})$.

The Howe duality conjecture for archimedean fields was proved by Roger Howe, and for p-adic fields with $p$ odd was proved by Jean-Loup Waldspurger. Wee Teck Gan and Shuichiro Takeda later gave a proof that works for any residue characteristic (excluding quaternionic dual pairs). The final case of quaternionic dual pairs (in any residual characteristic) was completed by Wee Teck Gan and Binyong Sun.

See also

• Reductive dual pair
• Metaplectic group

References

• Weil, André (1964), "Sur certains groupes d'opérateurs unitaires", Acta Math., 111: 143–211, doi:10.1007/BF02391012
• Howe, Roger E. (1979), "θ-series and invariant theory", in Borel, A.; 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)
• Waldspurger, Jean-Loup (1980), "Correspondance de Shimura", J. Math. Pures Appl., 59 (9): 1–132
• Rallis, Stephen (1984), "On the Howe duality conjecture", Compositio Math., 51: 333–399
• 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
• Howe, Roger E. (1989), "Transcending classical invariant theory", Journal of the American Mathematical Society, 2 (3), American Mathematical Society: 535–552, doi:10.2307/1990942, JSTOR 1990942.
• 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
• Waldspurger, Jean-Loup (1991), "Correspondances de Shimura et quaternions", Forum Math., 3 (3): 219–307, doi:10.1515/form.1991.3.219
• 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; Sun, Binyong (2017), "The Howe duality conjecture: quaternionic case", in Cogdell, J.; Kim, J.-L.; Zhu, C.-B. (eds.), Representation Theory, Number Theory, and Invariant Theory, Progr. Math., 323, Birkhäuser/Springer, pp. 175-192.