Theta correspondence

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In mathematics, the theta correspondence or Howe correspondence is a mathematical relation between representations of two groups of a reductive dual pair. The local theta correspondence relates irreducible admissible representations over a local field, while the global theta correspondence relates irreducible automorphic representations over a global field. The theta correspondence was introduced by Roger Howe in ( Howe 1979). Its 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 the theta correspondence.

Statement

Setup

Let [math]\displaystyle{ F }[/math] be a local or a global field, not of characteristic [math]\displaystyle{ 2 }[/math]. Let [math]\displaystyle{ W }[/math] be a symplectic vector space over [math]\displaystyle{ F }[/math], and [math]\displaystyle{ Sp(W) }[/math] the symplectic group.

Fix a reductive dual pair [math]\displaystyle{ (G,H) }[/math] in [math]\displaystyle{ Sp(W) }[/math]. There is a classification of reductive dual pairs.[1] [2]

Local theta correspondence

[math]\displaystyle{ F }[/math] is now a local field. Fix a non-trivial additive character [math]\displaystyle{ \psi }[/math] of [math]\displaystyle{ F }[/math]. There exists a Weil representation of the metaplectic group [math]\displaystyle{ Mp(W) }[/math] associated to [math]\displaystyle{ \psi }[/math], which we write as [math]\displaystyle{ \omega_{\psi} }[/math].

Given the reductive dual pair [math]\displaystyle{ (G,H) }[/math] in [math]\displaystyle{ Sp(W) }[/math], one obtains a pair of commuting subgroups [math]\displaystyle{ (\widetilde{G}, \widetilde{H}) }[/math] in [math]\displaystyle{ Mp(W) }[/math] by pulling back the projection map from [math]\displaystyle{ Mp(W) }[/math] to [math]\displaystyle{ Sp(W) }[/math].

The local theta correspondence is a 1-1 correspondence between certain irreducible admissible representations of [math]\displaystyle{ \widetilde{G} }[/math] and certain irreducible admissible representations of [math]\displaystyle{ \widetilde{H} }[/math], obtained by restricting the Weil representation [math]\displaystyle{ \omega_{\psi} }[/math] of [math]\displaystyle{ Mp(W) }[/math] to the subgroup [math]\displaystyle{ \widetilde{G}\cdot\widetilde{H} }[/math]. 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.

Key properties of local theta correspondence include its compatibility with Bernstein-Zelevinsky induction [3] and conservation relations concerning the first occurrence indices along Witt towers .[4]

Global theta correspondence

Stephen Rallis showed a version of the global Howe duality conjecture for cuspidal automorphic representations over a global field, assuming the validity of the Howe duality conjecture for all local places. [5]

Howe duality conjecture

Define [math]\displaystyle{ \mathcal{R}(\widetilde{G},\omega_{\psi}) }[/math] the set of irreducible admissible representations of [math]\displaystyle{ \widetilde{G} }[/math], which can be realized as quotients of [math]\displaystyle{ \omega_{\psi} }[/math]. Define [math]\displaystyle{ \mathcal{R}(\widetilde{H},\omega_{\psi}) }[/math] and [math]\displaystyle{ \mathcal{R}(\widetilde{G}\cdot\widetilde{H},\omega_{\psi}) }[/math], likewise.

The Howe duality conjecture asserts that [math]\displaystyle{ \mathcal{R}(\widetilde{G}\cdot\widetilde{H},\omega_{\psi}) }[/math] is the graph of a bijection between [math]\displaystyle{ \mathcal{R}(\widetilde{G},\omega_{\psi}) }[/math] and [math]\displaystyle{ \mathcal{R}(\widetilde{H},\omega_{\psi}) }[/math].

The Howe duality conjecture for archimedean local fields was proved by Roger Howe.[6] For [math]\displaystyle{ p }[/math]-adic local fields with [math]\displaystyle{ p }[/math] odd it was proved by Jean-Loup Waldspurger.[7] Alberto Mínguez later gave a proof for dual pairs of general linear groups, that works for arbitrary residue characteristic. [8] For orthogonal-symplectic or unitary dual pairs, it was proved by Wee Teck Gan and Shuichiro Takeda. [9] The final case of quaternionic dual pairs was completed by Wee Teck Gan and Binyong Sun.[10]

See also

References

  1. ↑ Howe 1979.
  2. ↑ Mœglin, Vignéras & Waldspurger 1987.
  3. ↑ Kudla 1986.
  4. ↑ Sun & Zhu 2015.
  5. ↑ Rallis 1984.
  6. ↑ Howe 1989.
  7. ↑ Waldspurger 1990.
  8. ↑ Mínguez 2008.
  9. ↑ Gan & Takeda 2016.
  10. ↑ Gan & Sun 2017.

Bibliography

  • Gan, Wee Teck; Takeda, Shuichiro (2016), "A proof of the Howe duality conjecture", J. Amer. Math. Soc. 29 (2): 473–493, doi:10.1090/jams/839 
  • Gan, Wee Teck; Sun, Binyong (2017), "The Howe duality conjecture: quaternionic case", in Cogdell, J.; Kim, J.-L.; Zhu, C.-B., Representation Theory, Number Theory, and Invariant Theory, Progr. Math., 323, Birkhäuser/Springer, pp. 175–192 
  • Howe, Roger E. (1979), "θ-series and invariant theory", in Borel, A.; Casselman, W., 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 
  • Howe, Roger E. (1989), "Transcending classical invariant theory", J. Amer. Math. Soc. 2 (3): 535–552, doi:10.2307/1990942 
  • Kudla, Stephen S. (1986), "On the local theta-correspondence", Invent. Math. 83 (2): 229–255, doi:10.1007/BF01388961 
  • Mínguez, Alberto (2008), "Correspondance de Howe explicite: paires duales de type II", Ann. Sci. Éc. Norm. Supér., 4 41 (5): 717–741, doi:10.24033/asens.2080 
  • 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 
  • Rallis, Stephen (1984), "On the Howe duality conjecture", Compositio Math. 51 (3): 333–399 
  • Sun, Binyong; Zhu, Chen-Bo (2015), "Conservation relations for local theta correspondence", J. Amer. Math. Soc. 28 (4): 939–983, doi:10.1090/S0894-0347-2014-00817-1 
  • Waldspurger, Jean-Loup (1980), "Correspondance de Shimura", J. Math. Pures Appl. 59 (9): 1–132 
  • 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 
  • Weil, André (1964), "Sur certains groupes d'opérateurs unitaires", Acta Math. 111: 143–211, doi:10.1007/BF02391012 



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