On Certain Classes of Unitary Representations for Split Classical Groups

Muić, Goran

Canadian journal of mathematics, Tome 59 (2007) no. 1, pp. 148-185

Résumé

In this paper we prove the unitarity of duals of tempered representations supported on minimal parabolic subgroups for split classical $p$ -adic groups. We also construct a family of unitary spherical representations for real and complex classical groups.

Détail

DOI : 10.4153/CJM-2007-007-0

Mots-clés : 22E35, 22E50, 11F70

Bibliographie

[A] Aubert, A. M., 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), 2179–2189 (and Erratum, Trans. Amer. Math. Soc 348(1996), 4687–4690). Google Scholar

[B] Ban, D., Parabolic inducition and Jacquet modules of representations of O(2n, F). Glas. Mat. Ser. III 34(54)(1999), 147–185. Google Scholar

[Ba] Barbasch, D., The unitary dual for complex classical Lie groups. Invent. Math. 96(1989), 103–176. Google Scholar

[BM1] Barbasch, D. and Moy, A., A unitary criterion for p-adic groups. Invent. Math. 98(1989), 19–37. Google Scholar

[BM2] Barbasch, D. and Moy, A., Reduction to real infinitesimal character in affine Hecke algebras. J. Amer. Math. Soc. 6(1993), 611–635. Google Scholar

[Car] Cartier, P. Representations of p-adic groups: a survey. In: Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence, RI, 1979, pp. 111–155. Google Scholar

[Jan1] Jantzen, C. Duality and supports for induced representations of orthogonal groups. Canad. J. Math. 57(2005), no. 1, 159–179. Google Scholar

[Jan2] Jantzen, C., Jacquet modules of p-adic general linear groups. Preprint http://personal.ecu.edu/jantzenc/driverGL.pdf Google Scholar

[JK] Jantzen, C. and Kim, H., Parametrization of the image of normalized intertwining operators. Pacific J. Math. 199(2001), 367–415. Google Scholar

[K1] Kim, H., The residual spectrum of G . Canad. J. Math. 48(1996), 1245–1272. Google Scholar

[K2] Kim, H., Residual spectrum of odd orthogonal groups. Internat. Math. Res. Notices 17(2001), 873–906. Google Scholar

[KR] Kudla, S. S. and Rallis, S., Ramified degenerate principal series representations for Sp(n). Israel J. Math. 78(1992), 209–256. Google Scholar

[KSh] Keys, K. and Shahidi, F., Artin L-functions and normalization of intertwining operators. Ann. Sci. École Norm. Sup. 21(1988), 67–89. Google Scholar

[La] Langlands, R. P., On the notion of an automorphic representation. In: Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence, RI, 1979, pp. 203–207. (This is a supplement to: Borel, A. and Jacquet, H., Automorphic forms and automorphic representations. pp. 189–202.) Google Scholar

[LMT] Lapid, E., Muić, G., and Tadić, M., On the generic unitary dual of quasisplit classical groups. Intnat. Math. Res. Notices (2004), no. 26, 1335–1354. Google Scholar

[Mœ1] Mœglin, C., Orbites unipotentes et spectre discret non ramifié: le cas des groupes classiques déployés. Compositio Math. 77(1991), 1–54. Google Scholar

[Mœ2] Lapid, E., Muić, G., and Tadić, M., Représentations unipotentes et formes automorphes de carré intégrable. Forum Math. 6(1994), 651–744. Google Scholar

[Mœ3] Lapid, E., Muić, G., and Tadić, M., Sur la classification des séries discrètes des groupes classiques p-adiques: paramètres de Langlands et exhaustivité. J. Eur. Math. Soc. (JEMS) 4(2002), 143–200. Google Scholar

[MT] Mœglin, C. and Tadić, M., Construction of discrete series for classical p-adic groups. J. Amer. Math. Soc. 15(2002), 715–786. Google Scholar

[MW1] Mœglin, C. and Waldspurger, J.-L., Sur l’ involution de Zelevinski. J. Reine Angew. Math. 372(1986), 136–177. Google Scholar

[MW2] Mœglin, C. and Waldspurger, J.-L., Le spectre résiduel de GL(n). Ann. Sci. École Norm. Sup. 22(1989), 605–674. Google Scholar

[MW3] Mœglin, C. and Waldspurger, J.-L., Spectral Decomposition and Eisenstein Series. Une paraphrase de l’criture. Cambridge Tracts in Mathematics 113, Cambridge University Press, Cambridge, 1995. Google Scholar

[MVW] Moeglin, C., Vignéras, M.-F., and Waldspurger, J.-L., Correspondence de Howe sur un corps p-adique. Lecture Notes in Mathematics 1291, Springer-Verlag, Berlin, 1987. Google Scholar

[M1] Muić, G., The unitary dual of p-adic G . Duke Math. J. 90(1997), 465–493. Google Scholar

[M2] Muić, G., Some results on square integrable representations; Irreducibility of standard representations. Internat. Math. Res. Notices (1998), no. 14, 705–726. Google Scholar

[M3] Muić, G., Reducibility of generalized principal series. Canad. J. Math. 57(2005), 616–647. Google Scholar

[M4] Muić, G., On the non-unitary unramified dual for classical p-adic groups. Trans. Amer.Math. Soc 350(2006), 4653–4687. Google Scholar

[SS] Schneider, P. and Stuhler, U., Representation theory and sheaves on the Bruhat–Tits building. Inst. Hautes Études Sci. Publ. Math. 85(1997), 97–191. Google Scholar

[Sh1] Shahidi, F., Fourier transforms of intertwining operators and Plancherel measures for GL(n). Amer. J. Math. 106(1984), 67–111. Google Scholar

[Sh2] Shahidi, F., A proof of Langland's conjecture on Plancherel measures; complementary series for p-adic groups. Ann. of Math. 132(1990), 273–330. Google Scholar

[Sh3] Shahidi, F., Twisted endoscopy and reducibility of induced representations for p-adic groups. Duke Math. J. 66(1992), 1–41. Google Scholar

[Sp] Speh, B., Unitary representations of GL(n, ℝ) with nontrivial (g, K)-cohomology. Invent. Math. 71(1983), 443–465. Google Scholar

[T1] Tadić, M., Classification of unitary representations in irreducible representations of general linear group (non-Archimedean case). Ann. Sci. École Norm. Sup. 19(1986), 335–382. Google Scholar

[T2] Tadić, M., An external approach to unitary representations. Bull. Amer. Math. Soc. 28(1993), 215–252. Google Scholar

[T3] Tadić, M., Structure arising from induction and Jacquet modules of representations of classical p–adic groups. J. Algebra 177(1995), 1–33. Google Scholar

[Ta] Tate, J., Number theoretic background. In: Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math. 33, American Mathematical Society, Providence, RI, 1979, pp. 3–26. Google Scholar

[W] Waldspurger, J.-L., La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra. J. Inst. Math. Jussieu 2(2003), 235–333. Google Scholar

[Ža] Žampera, S., The residual spectrum of the group of type G . J. Math. Pures Appl. 76(1997), 805–835. Google Scholar

[Ze] Zelevinsky, A. V., Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. 13(1980), 165–210. Google Scholar