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Explicit Howe correspondence: Dual pairs of type II. (Correspondance de Howe explicite: Paires duales de type II.) (French) Zbl 1220.22014
Let $$F$$ be a nonarchimedean local field. This paper establishes the bijectivity of the Howe correspondence in the Type II case, that is, when the dual pair is of the form $$(GL_n(D), GL_m(D))$$ for a division algebra $$D$$ with center $$F$$. The case of Type I, meaning a symplectic-orthogonal pair, was established in [J.-L. Waldspurger, Isr. Math. Conf. Proc. 2, 267–324 (1990; Zbl 0722.22009)] and [C. Moeglin, M.-F. Vignéras and J.-L. Waldspurger, Correspondances de Howe sur un corps $$p$$-adique. Lecture Notes in Mathematics, 1291. Subseries: Mathematisches Institut der Universität und Max-Planck-Institut für Mathematik, Bonn, Vol. 11. Berlin etc.: Springer-Verlag. (1987; Zbl 0642.22002)]. The bijectivity in the present case of Type II had previously been done in private notes of Howe, by a different method.
The present method uses an adaptation of the filtration method of S. S. Kudla [Invent. Math. 83, 229–255 (1986; Zbl 0583.22010)], as well as previous work of the author [J. Reine Angew. Math. 629, 107–131 (2009; Zbl 1172.22008)] concerning uniqueness of irreducible quotients of induced representations in the $$GL_n(D)$$ case. The bijection is given explicitly in terms of Langlands parameters.

##### MSC:
 2.2e+51 Representations of Lie and linear algebraic groups over local fields
##### Keywords:
theta correspondence; dual pairs; Langlands parameters
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