Duality for classical 𝑝-adic groups: The half-integral case

Jantzen, Chris

doi:10.1090/ert/519

Duality for classical $p$-adic groups: The half-integral case
HTML articles powered by AMS MathViewer

by Chris Jantzen PDF

Represent. Theory 22 (2018), 160-201 Request permission

Abstract:

Let $G$ be a classical $p$-adic group and let $\pi$ be a smooth irreducible representation of $G$. In this paper, we consider the problem of calculating the dual (in the sense of Aubert and Schneider-Stuhler) $\hat {\pi }$. More precisely, if $\pi$ is specified by its Langlands data, the problem is to determine the Langlands data for $\hat {\pi }$. This problem reduces (based on supercuspidal support) to two main cases: half-integral reducibility and integral reducibility; the latter is addressed here.

References

Dean Alvis, The duality operation in the character ring of a finite Chevalley group, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 6, 907–911. MR 546315, DOI 10.1090/S0273-0979-1979-14690-1
James Arthur, The endoscopic classification of representations, American Mathematical Society Colloquium Publications, vol. 61, American Mathematical Society, Providence, RI, 2013. Orthogonal and symplectic groups. MR 3135650, DOI 10.1090/coll/061
Anne-Marie Aubert, Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif $p$-adique, Trans. Amer. Math. Soc. 347 (1995), no. 6, 2179–2189 (French, with English summary). MR 1285969, DOI 10.1090/S0002-9947-1995-1285969-0
Dubravka Ban, Parabolic induction and Jacquet modules of representations of $\textrm {O}(2n,F)$, Glas. Mat. Ser. III 34(54) (1999), no. 2, 147–185. MR 1739616
Dubravka Ban and Chris Jantzen, Degenerate principal series for even-orthogonal groups, Represent. Theory 7 (2003), 440–480. MR 2017065, DOI 10.1090/S1088-4165-03-00166-3
Dubravka Ban and Chris Jantzen, Duality and the normalization of standard intertwining operators, Manuscripta Math. 115 (2004), no. 4, 401–415. MR 2103658, DOI 10.1007/s00229-004-0504-7
Dubravka Ban and Chris Jantzen, Jacquet modules and the Langlands classification, Michigan Math. J. 56 (2008), no. 3, 637–653. MR 2490651, DOI 10.1307/mmj/1231770365
Dan Barbasch and Allen Moy, A unitarity criterion for $p$-adic groups, Invent. Math. 98 (1989), no. 1, 19–37. MR 1010153, DOI 10.1007/BF01388842
I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), no. 4, 441–472. MR 579172, DOI 10.24033/asens.1333
Armand Borel and Nolan R. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Annals of Mathematics Studies, No. 94, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 554917
W. Casselman, Introduction to the theory of admissible representations of $p$-adic reductive groups, preprint (available online at www.math.ubc.ca/people/faculty/cass/research.html as “The $p$-adic notes”).
Charles W. Curtis, Truncation and duality in the character ring of a finite group of Lie type, J. Algebra 62 (1980), no. 2, 320–332. MR 563231, DOI 10.1016/0021-8693(80)90185-4
Pierre Deligne and George Lusztig, Duality for representations of a reductive group over a finite field, J. Algebra 74 (1982), no. 1, 284–291. MR 644236, DOI 10.1016/0021-8693(82)90023-0
Pierre Deligne and George Lusztig, Duality for representations of a reductive group over a finite field. II, J. Algebra 81 (1983), no. 2, 540–545. MR 700298, DOI 10.1016/0021-8693(83)90202-8
W.-T. Gan and L. Lomelí, Globalization of supercuspidal representations over function fields and applications, to appear, J. Eur. Math. Soc.
N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR 185016, DOI 10.1007/BF02684396
Chris Jantzen, Reducibility of certain representations for symplectic and odd-orthogonal groups, Compositio Math. 104 (1996), no. 1, 55–63. MR 1420710
Chris Jantzen, On supports of induced representations for symplectic and odd-orthogonal groups, Amer. J. Math. 119 (1997), no. 6, 1213–1262. MR 1481814, DOI 10.1353/ajm.1997.0039
Chris Jantzen, Some remarks on degenerate principal series, Pacific J. Math. 186 (1998), no. 1, 67–87. MR 1665057, DOI 10.2140/pjm.1998.186.67
Chris Jantzen, Duality and supports of induced representations for orthogonal groups, Canad. J. Math. 57 (2005), no. 1, 159–179. MR 2113853, DOI 10.4153/CJM-2005-007-x
Chris Jantzen, Jacquet modules of $p$-adic general linear groups, Represent. Theory 11 (2007), 45–83. MR 2306606, DOI 10.1090/S1088-4165-07-00316-0
Chris Jantzen, Discrete series for $p$-adic $\textrm {SO}(2n)$ and restrictions of representations of $\textrm {O}(2n)$, Canad. J. Math. 63 (2011), no. 2, 327–380. MR 2809059, DOI 10.4153/CJM-2011-003-2
Chris Jantzen, Tempered representations for classical $p$-adic groups, Manuscripta Math. 145 (2014), no. 3-4, 319–387. MR 3268853, DOI 10.1007/s00229-014-0679-5
Chris Jantzen, Jacquet modules and irrreducibility of induced representations for classical $p$-adic groups, Manuscripta Math. 156 (2018), no. 1-2, 23–55. MR 3783564, DOI 10.1007/s00229-017-0955-2
Shin-ichi Kato, Duality for representations of a Hecke algebra, Proc. Amer. Math. Soc. 119 (1993), no. 3, 941–946. MR 1215028, DOI 10.1090/S0002-9939-1993-1215028-8
N. Kawanaka, Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field, Invent. Math. 69 (1982), no. 3, 411–435. MR 679766, DOI 10.1007/BF01389363
Harold Knight and Andrei Zelevinsky, Representations of quivers of type $A$ and the multisegment duality, Adv. Math. 117 (1996), no. 2, 273–293. MR 1371654, DOI 10.1006/aima.1996.0013
Takuya Konno, A note on the Langlands classification and irreducibility of induced representations of $p$-adic groups, Kyushu J. Math. 57 (2003), no. 2, 383–409. MR 2050093, DOI 10.2206/kyushujm.57.383
C. Mœglin, Normalisation des opérateurs d’entrelacement et réductibilité des induites des cuspidales; le cas des groupes classiques $p$-adiques, Ann. of Math., 151(2000), 817-847.
C. Mœglin, Sur la classification des séries discrètes des groupes classiques $p$-adiques: paramètres de Langlands et exhaustivité, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 2, 143–200 (French, with English summary). MR 1913095, DOI 10.1007/s100970100033
C. Mœglin, Sur certains paquets d’Arthur et involution d’Aubert-Schneider-Stuhler généralisée, Represent. Theory 10 (2006), 86–129 (French, with English summary). MR 2209850, DOI 10.1090/S1088-4165-06-00270-6
Colette Mœglin, Paquets stables des séries discrètes accessibles par endoscopie tordue; leur paramètre de Langlands, Automorphic forms and related geometry: assessing the legacy of I. I. Piatetski-Shapiro, Contemp. Math., vol. 614, Amer. Math. Soc., Providence, RI, 2014, pp. 295–336 (French, with English summary). MR 3220932, DOI 10.1090/conm/614/12254
Colette Mœglin and Marko Tadić, Construction of discrete series for classical $p$-adic groups, J. Amer. Math. Soc. 15 (2002), no. 3, 715–786. MR 1896238, DOI 10.1090/S0894-0347-02-00389-2
C. Mœglin and J.-L. Waldspurger, Sur l’involution de Zelevinski, J. Reine Angew. Math. 372 (1986), 136–177 (French). MR 863522, DOI 10.1515/crll.1986.372.136
Goran Muić, The unitary dual of $p$-adic $G_2$, Duke Math. J. 90 (1997), no. 3, 465–493. MR 1480543, DOI 10.1215/S0012-7094-97-09012-8
Goran Muić, Composition series of generalized principal series; the case of strongly positive discrete series, Israel J. Math. 140 (2004), 157–202. MR 2054843, DOI 10.1007/BF02786631
Goran Muić, Reducibility of generalized principal series, Canad. J. Math. 57 (2005), no. 3, 616–647. MR 2134404, DOI 10.4153/CJM-2005-025-4
Goran Muić and Marko Tadić, Unramified unitary duals for split classical $p$-adic groups; the topology and isolated representations, On certain $L$-functions, Clay Math. Proc., vol. 13, Amer. Math. Soc., Providence, RI, 2011, pp. 375–438. MR 2767523
Peter Schneider and Ulrich Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Inst. Hautes Études Sci. Publ. Math. 85 (1997), 97–191. MR 1471867, DOI 10.1007/BF02699536
Freydoon Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for $p$-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. MR 1070599, DOI 10.2307/1971524
Freydoon Shahidi, Twisted endoscopy and reducibility of induced representations for $p$-adic groups, Duke Math. J. 66 (1992), no. 1, 1–41. MR 1159430, DOI 10.1215/S0012-7094-92-06601-4
Allan J. Silberger, The Langlands quotient theorem for $p$-adic groups, Math. Ann. 236 (1978), no. 2, 95–104. MR 507262, DOI 10.1007/BF01351383
Allan J. Silberger, Special representations of reductive $p$-adic groups are not integrable, Ann. of Math. (2) 111 (1980), no. 3, 571–587. MR 577138, DOI 10.2307/1971110
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
Marko Tadić, On reducibility of parabolic induction, Israel J. Math. 107 (1998), 29–91. MR 1658535, DOI 10.1007/BF02764004
Marko Tadić, On classification of some classes of irreducible representations of classical groups, Representations of real and $p$-adic groups, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., vol. 2, Singapore Univ. Press, Singapore, 2004, pp. 95–162. MR 2090870, DOI 10.1142/9789812562500_{0}004
Marko Tadić, On invariants of discrete series representations of classical $p$-adic groups, Manuscripta Math. 135 (2011), no. 3-4, 417–435. MR 2813443, DOI 10.1007/s00229-010-0423-8
Marko Tadić, On tempered and square integrable representations of classical $p$-adic groups, Sci. China Math. 56 (2013), no. 11, 2273–2313. MR 3123571, DOI 10.1007/s11425-013-4667-0
J.-L. Waldspurger, La formule de Plancherel pour les groupes $p$-adiques (d’après Harish-Chandra), J. Inst. Math. Jussieu 2 (2003), no. 2, 235–333 (French, with French summary). MR 1989693, DOI 10.1017/S1474748003000082
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
Yuanli Zhang, $L$-packets and reducibilities, J. Reine Angew. Math. 510 (1999), 83–102. MR 1696092, DOI 10.1515/crll.1999.050