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). Theta correspondence is a generalisation of Shimura correspondence. It's also named as Howe conjecture. Theta correspondence is a correspondence between certain representations on Mp(2n) and these on SO(2n+1). In fact, it's not yet fully constructed. The case n = 1 is already completely proved by J. L. Waldspurger in Waldspurger (1980) and Waldspurger (1991).

Statement

Let E be a nonarchimedean local field of characteristic not 2, with its quotient field of charcateristic p. Let F be a quadratic extension over E. Let V ( resp. W ) be a n-dimensional hermitian space ( resp. a 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 ψ of F for the pair G(V)×H(W), which we write as ρ(ψ). Let π be a irreducible admissible representation of G(V). Here, we only consider the case G(V)×H(W) = SO(n)×SO(m) or U(n)×U(m). We can find a certain representation θ(π,ψ) of H(W), which is in fact a certain quotient of the Weil representation ρ(ψ) by π. Now we are ready to state the conjecture of Howe.

Howe duality Conjecture.

(i) θ(π,ψ) is irreducible or 0;

(ii) Let π, π' be two irreducible admissible representations of G(V), s. t. θ(π,ψ) = θ(π',ψ) ≠ 0. Then, π = π'.

Here, we have used the notations of Gan & Takeda (2014). The Howe conjecture in our setting was already prove by J. L. Waldspurger in Waldspurger (1990). W. T. Gan and S. Takeda reprove it, using a simpler and more uniform method, in Gan & Takeda (2014). Thanks to their works, we are able to define the local theta correspondence in our situation.

Definition. ( local theta correspondence ). Let Irr(G(V)) ( resp. Irr(H(W)) ) be the set of all irreducible admissible representations of G(V) ( resp. H(W) ). Let θ be the map Irr(G(V)) → Irr(H(W)), which associates every irreducible admissible representation π of G(V) the irreducible admissible representation θ(π,ψ) of H(W). We call θ the local theta correspondence for the pair G(V)×H(W).

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 G(V) as well, see Waldspurger (1991).

Reason for the name

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

See also

References

  • Howe, Roger (1979), "θ-series and invariant theory" (PDF), in Borel, Armand; 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, 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
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