Construction of discrete series for classical 𝑝-adic groups

Mœglin, Colette; Tadić, Marko

doi:10.1090/S0894-0347-02-00389-2

Construction of discrete series for classical $p$-adic groups
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by Colette Mœglin and Marko Tadić;

J. Amer. Math. Soc. 15 (2002), 715-786

DOI: https://doi.org/10.1090/S0894-0347-02-00389-2

Published electronically: April 5, 2002

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Abstract:

The classification of irreducible square integrable representations of classical $p$-adic groups is completed in this paper, under a natural local assumption. Further, this classification gives a parameterization of irreducible tempered representations of these groups. Therefore, it implies a classification of the non-unitary duals of these groups (modulo cuspidal data). The classification of irreducible square integrable representations is directly related to the parameterization of irreducible square integrable representations in terms of dual objects, which is predicted by Langlands program.

References

J. Adams, $L$-functoriality for dual pairs, Astérisque 171-172 (1989), 85–129. Orbites unipotentes et représentations, II. MR 1021501
James Arthur, Unipotent automorphic representations: global motivation, Automorphic forms, Shimura varieties, and $L$-functions, Vol. I (Ann Arbor, MI, 1988) Perspect. Math., vol. 10, Academic Press, Boston, MA, 1990, pp. 1–75. MR 1044818
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
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
David Goldberg, Reducibility of induced representations for $\textrm {Sp}(2n)$ and $\textrm {SO}(n)$, Amer. J. Math. 116 (1994), no. 5, 1101–1151. MR 1296726, DOI 10.2307/2374942
H. Jacquet, I. I. Piatetskii-Shapiro, and J. A. Shalika, Rankin-Selberg convolutions, Amer. J. Math. 105 (1983), no. 2, 367–464. MR 701565, DOI 10.2307/2374264
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, On square-integrable representations of classical $p$-adic groups. II, Represent. Theory 4 (2000), 127–180. MR 1789464, DOI 10.1090/S1088-4165-00-00081-9
David Kazhdan and George Lusztig, Proof of the Deligne-Langlands conjecture for Hecke algebras, Invent. Math. 87 (1987), no. 1, 153–215. MR 862716, DOI 10.1007/BF01389157
Stephen S. Kudla and Stephen Rallis, A regularized Siegel-Weil formula: the first term identity, Ann. of Math. (2) 140 (1994), no. 1, 1–80. MR 1289491, DOI 10.2307/2118540
C. Mœglin, Normalisation des opérateurs d’entrelacement et réductibilité des induites de cuspidales; le cas des groupes classiques $p$-adiques, Ann. of Math. (2) 151 (2000), no. 2, 817–847 (French). MR 1765711, DOI 10.2307/121049

Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité

C. Mœglin, Représentations quadratiques unipotentes des groupes classiques $p$-adiques, Duke Math. J. 84 (1996), no. 2, 267–332 (French). MR 1404331, DOI 10.1215/S0012-7094-96-08410-0
Colette Mœglin, Non nullité de certains relêvements par séries théta, J. Lie Theory 7 (1997), no. 2, 201–229 (French, with English summary). MR 1473165

Points de réductibilité pour les induites de cuspidales

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
C. Mœglin and J.-L. Waldspurger, Le spectre résiduel de $\textrm {GL}(n)$, Ann. Sci. École Norm. Sup. (4) 22 (1989), no. 4, 605–674 (French). MR 1026752, DOI 10.24033/asens.1595
Goran Muić, On generic irreducible representations of $\textrm {Sp}(n,F)$ and $\textrm {SO}(2n+1,F)$, Glas. Mat. Ser. III 33(53) (1998), no. 1, 19–31. MR 1652772
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, On certain $L$-functions, Amer. J. Math. 103 (1981), no. 2, 297–355. MR 610479, DOI 10.2307/2374219
Freydoon Shahidi, Local coefficients and normalization of intertwining operators for $\textrm {GL}(n)$, Compositio Math. 48 (1983), no. 3, 271–295. MR 700741
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
Freydoon Shahidi, Fourier transforms of intertwining operators and Plancherel measures for $\textrm {GL}(n)$, Amer. J. Math. 106 (1984), no. 1, 67–111. MR 729755, DOI 10.2307/2374430
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, On regular square integrable representations of $p$-adic groups, Amer. J. Math. 120 (1998), no. 1, 159–210. MR 1600280, DOI 10.1353/ajm.1998.0007
Marko Tadić, On reducibility of parabolic induction, Israel J. Math. 107 (1998), 29–91. MR 1658535, DOI 10.1007/BF02764004
Marko Tadić, Square integrable representations of classical $p$-adic groups corresponding to segments, Represent. Theory 3 (1999), 58–89. MR 1698200, DOI 10.1090/S1088-4165-99-00071-0

A family of square integrable representations of classical $p$-adic groups

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, Representations of $p$-adic symplectic groups, Compositio Math. 90 (1994), no. 2, 123–181. MR 1266251
David A. Vogan Jr., The local Langlands conjecture, Representation theory of groups and algebras, Contemp. Math., vol. 145, Amer. Math. Soc., Providence, RI, 1993, pp. 305–379. MR 1216197, DOI 10.1090/conm/145/1216197

La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra

J.-L. Waldspurger, Un exercice sur $\textrm {GSp}(4,F)$ et les représentations de Weil, Bull. Soc. Math. France 115 (1987), no. 1, 35–69 (French). MR 897614, DOI 10.24033/bsmf.2068
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