A proof of the Howe duality conjecture

Gan, Wee Teck ¹   ; Takeda, Shuichiro ²

¹ Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076
² Mathematics Department, University of Missouri, 202 Math Sciences Building, Columbia, Missouri 65211

Journal of the American Mathematical Society, Tome 29 (2016) no. 2, pp. 473-493

Résumé

We give a proof of the Howe duality conjecture in the theory of local theta correspondence for symplectic-orthogonal or unitary dual pairs in arbitrary residual characteristic.

Détail

DOI : 10.1090/jams/839

Bibliographie

[1] Gan, Wee Teck, Ichino, Atsushi Formal degrees and local theta correspondence Invent. Math. 2014 509 672

[2] Gan, Wee Teck, Savin, Gordan Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence Compos. Math. 2012 1655 1694

[3] Howe, R. 𝜃-series and invariant theory 1979 275 285

[4] Howe, Roger Transcending classical invariant theory J. Amer. Math. Soc. 1989 535 552

[5] Hanzer, M. Inducirane reprezentacije hermitskih kvaternionskih grupa

[6] Hanzer, Marcela, Muić, Goran Parabolic induction and Jacquet functors for metaplectic groups J. Algebra 2010 241 260

[7] Kudla, Stephen S. On the local theta-correspondence Invent. Math. 1986 229 255

[8] Kudla, Stephen S. Splitting metaplectic covers of dual reductive pairs Israel J. Math. 1994 361 401

[9] Kudla, Stephen S., Rallis, Stephen On first occurrence in the local theta correspondence 2005 273 308

[10] Li, Jian-Shu, Sun, Binyong, Tian, Ye The multiplicity one conjecture for local theta correspondences Invent. Math. 2011 117 124

[11] Lin, Yanan, Sun, Binyong, Tan, Shaobin MVW-extensions of quaternionic classical groups Math. Z. 2014 81 89

[12] Mínguez, Alberto Correspondance de Howe explicite: paires duales de type II Ann. Sci. Éc. Norm. Supér. (4) 2008 717 741

[13] Mœglin, Colette, Vignéras, Marie-France, Waldspurger, Jean-Loup Correspondances de Howe sur un corps 𝑝-adique 1987

[14] Muić, Goran Howe correspondence for discrete series representations J. Reine Angew. Math. 2004 99 150

[15] Muić, Goran On the structure of the full lift for the Howe correspondence of (𝑆𝑝(𝑛),𝑂(𝑉)) for rank-one reducibilities Canad. Math. Bull. 2006 578 591

[16] Muić, Goran On the structure of theta lifts of discrete series for dual pairs (𝑆𝑝(𝑛),𝑂(𝑉)) Israel J. Math. 2008 87 124

[17] Muić, Goran Theta lifts of tempered representations for dual pairs (𝑆𝑝_{2𝑛},𝑂(𝑉)) Canad. J. Math. 2008 1306 1335

[18] Tadić, Marko Structure arising from induction and Jacquet modules of representations of classical 𝑝-adic groups J. Algebra 1995 1 33

[19] Waldspurger, J.-L. Démonstration d’une conjecture de dualité de Howe dans le cas 𝑝-adique, 𝑝≠2 1990 267 324

[20] Zelevinsky, A. V. Induced representations of reductive 𝔭-adic groups. II. On irreducible representations of 𝔊𝔏(𝔫) Ann. Sci. École Norm. Sup. (4) 1980 165 210