A proof of the Howe duality conjecture

Gan, Wee Teck; Takeda, Shuichiro

doi:10.1090/jams/839

A proof of the Howe duality conjecture
HTML articles powered by AMS MathViewer

by Wee Teck Gan and Shuichiro Takeda

J. Amer. Math. Soc. 29 (2016), 473-493

DOI: https://doi.org/10.1090/jams/839

Published electronically: July 13, 2015

PDF | Request permission

Previous version: Original version posted July 13, 2015
Corrected version: Current version corrects publisher's error which provided incorrect information for bibliographic reference [GT].

Abstract:

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.

References

Wee Teck Gan and Atsushi Ichino, Formal degrees and local theta correspondence, Invent. Math. 195 (2014), no. 3, 509–672. MR 3166215, DOI 10.1007/s00222-013-0460-5
Wee Teck Gan and Gordan Savin, Representations of metaplectic groups I: epsilon dichotomy and local Langlands correspondence, Compos. Math. 148 (2012), no. 6, 1655–1694. MR 2999299, DOI 10.1112/S0010437X12000486
W. T. Gan and S. Takeda, On the Howe duality conjecture in classical theta correspondence, to appear in Contemporary Math., volume in honor of J. Cogdell (2014), arXiv:1405.2626.
R. Howe, $\theta$-series and invariant theory, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 275–285. MR 546602
Roger Howe, Transcending classical invariant theory, J. Amer. Math. Soc. 2 (1989), no. 3, 535–552. MR 985172, DOI 10.1090/S0894-0347-1989-0985172-6
M. Hanzer, Inducirane reprezentacije hermitskih kvaternionskih grupa. Ph.D. Thesis (2005), University of Zagreb.
Marcela Hanzer and Goran Muić, Parabolic induction and Jacquet functors for metaplectic groups, J. Algebra 323 (2010), no. 1, 241–260. MR 2564837, DOI 10.1016/j.jalgebra.2009.07.001
Stephen S. Kudla, On the local theta-correspondence, Invent. Math. 83 (1986), no. 2, 229–255. MR 818351, DOI 10.1007/BF01388961
Stephen S. Kudla, Splitting metaplectic covers of dual reductive pairs, Israel J. Math. 87 (1994), no. 1-3, 361–401. MR 1286835, DOI 10.1007/BF02773003
Stephen S. Kudla and Stephen Rallis, On first occurrence in the local theta correspondence, Automorphic representations, $L$-functions and applications: progress and prospects, Ohio State Univ. Math. Res. Inst. Publ., vol. 11, de Gruyter, Berlin, 2005, pp. 273–308. MR 2192827, DOI 10.1515/9783110892703.273
Jian-Shu Li, Binyong Sun, and Ye Tian, The multiplicity one conjecture for local theta correspondences, Invent. Math. 184 (2011), no. 1, 117–124. MR 2782253, DOI 10.1007/s00222-010-0287-2
Yanan Lin, Binyong Sun, and Shaobin Tan, MVW-extensions of quaternionic classical groups, Math. Z. 277 (2014), no. 1-2, 81–89. MR 3205764, DOI 10.1007/s00209-013-1246-6
Alberto Mínguez, Correspondance de Howe explicite: paires duales de type II, Ann. Sci. Éc. Norm. Supér. (4) 41 (2008), no. 5, 717–741 (French, with English and French summaries). MR 2504432, DOI 10.24033/asens.2080
Colette Mœglin, Marie-France Vignéras, and Jean-Loup Waldspurger, Correspondances de Howe sur un corps $p$-adique, Lecture Notes in Mathematics, vol. 1291, Springer-Verlag, Berlin, 1987 (French). MR 1041060, DOI 10.1007/BFb0082712
Goran Muić, Howe correspondence for discrete series representations; the case of $(\textrm {Sp}(n),\textrm {O}(V))$, J. Reine Angew. Math. 567 (2004), 99–150. MR 2038306, DOI 10.1515/crll.2004.014
Goran Muić, On the structure of the full lift for the Howe correspondence of $(\textrm {Sp}(n),\textrm {O}(V))$ for rank-one reducibilities, Canad. Math. Bull. 49 (2006), no. 4, 578–591. MR 2269768, DOI 10.4153/CMB-2006-054-3
Goran Muić, On the structure of theta lifts of discrete series for dual pairs $(\textrm {Sp}(n),\textrm {O}(V))$, Israel J. Math. 164 (2008), 87–124. MR 2391142, DOI 10.1007/s11856-008-0022-5
Goran Muić, Theta lifts of tempered representations for dual pairs $(\textrm {Sp}_{2n},\textrm {O}(V))$, Canad. J. Math. 60 (2008), no. 6, 1306–1335. MR 2462449, DOI 10.4153/CJM-2008-056-6
Marko Tadić, Structure arising from induction and Jacquet modules of representations of classical $p$-adic groups, J. Algebra 177 (1995), no. 1, 1–33. MR 1356358, DOI 10.1006/jabr.1995.1284
J.-L. Waldspurger, Démonstration d’une conjecture de dualité de Howe dans le cas $p$-adique, $p\neq 2$, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part I (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 2, Weizmann, Jerusalem, 1990, pp. 267–324 (French). MR 1159105
A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. II. On irreducible representations of $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 2, 165–210. MR 584084, DOI 10.24033/asens.1379