Duality and Supports of Induced Representations for Orthogonal Groups

Jantzen, Chris

Canadian journal of mathematics, Tome 57 (2005) no. 1, pp. 159-179

Résumé

In this paper, we construct a duality for $p$ -adic orthogonal groups.

Détail

DOI : 10.4153/CJM-2005-007-x

Mots-clés : 22E50

Bibliographie

[Al] [Al] Alvis, D., The duality operator in the character ring of a finite Chevalley group. Bull. Amer. Math. Soc. 1 (1979), 907–911. Google Scholar

[Au1] [Au1] 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. Google Scholar

[Au2] [Au2] Aubert, A.-M., Erratum: “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. 348 (1996), 4687–4690. Google Scholar

[Ba1] [Ba1] Ban, D., Parabolic induction and Jacquet modules of representations of O(2n, F). Glas. Mat. Ser. III 34(54)(1999), 147–185. Google Scholar

[Ba2] [Ba2] Ban, D., The Aubert involution and R-groups. Ann. Sci. École Norm. Sup. 35 (2002), 673–693. Google Scholar

[Ba3] [Ba3] Ban, D., Linear independence of intertwining operators. J. Algebra 271 (2004), 749–767. Google Scholar

[B-J1] [B-J1] Ban, D. and Jantzen, C., The Langlands classification for non-connected p-adic groups. Israel J. Math. 126 (2001), 239–261. Google Scholar

[B-J2] [B-J2] Ban, D. and Jantzen, C., Degenerate principal series for even orthogonal groups. Represent. Theory 7 (2003), 440–480. Google Scholar

[B-J3] [B-J3] Ban, D. and Jantzen, C., The Langlands classification for non-connected p-adic groups II: Multiplicity one. Proc. Amer.Math. Soc. 131 (2003), 3297–3304. Google Scholar

[Ba-Zh] [Ba-Zh] Ban, D. and Zhang, Y., Arthur R-groups, classical R-groups, and Aubert involutions for SO(2n+1), preprint. Google Scholar

[BDK] [BDK] Bernstein, J., Deligne, P. and Kazhdan, D., Trace Paley-Wiener theorem for reductive p-adic groups. J. AnalyseMath. 47 (1986), 180–192. Google Scholar

[B-Z] [B-Z] Bernstein, I. and Zelevinsky, A., Induced representations of reductive p-adic groups I. Ann. Sci. É cole Norm. Sup. 10 (1977), 441–472. Google Scholar

[Bo] [Bo] Borel, A., Linear algebraic groups. In: Algebraic groups and discontinuous subgroups. Amer. Math. Soc., Providence, RI, 1966, pp. 3–19. Google Scholar

[B-W] [B-W] Borel, A. and Wallach, N., Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups. Princeton University Press, Princeton, NJ, 1980. Google Scholar

[Br] [Br] Bruhat, F., Lectures on Some Aspects of p-adic Analysis. Tata Institute for Fundamental Research, Bombay, 1963. Google Scholar

[Ca] [Ca] Casselman, W., Introduction to the theory of admissible representations of p-adic reductive groups. preprint. Google Scholar

[Cu] [Cu] Curtis, C., Truncation and duality in the character ring of a finite group of Lie type. J. Algebra, 62 (1980), 320–332. Google Scholar

[D-L1] [D-L1] Deligne, P. and Lusztig, G., Duality for representations of reductive groups over a finite field. J. Algebra, 74 (1982), 284–291. Google Scholar

[D-L2] [D-L2] Deligne, P. and Lusztig, G., Duality for representations of reductive groups over a finite field, II. J. Algebra, 81 (1983), 540–545. Google Scholar

[D-M] [D-M] Digne, F. and Michel, J., Groupes réductifs non connexes, Ann. Sci. École Norm. Sup. 27 (1994), 345–406. Google Scholar

[G-K] [G-K] Gelbart, S. and Knapp, A., L-indistinguishability and R groups for the special linear group. Adv. in Math. 43 (1982), 101–121. Google Scholar

[Go1] [Go1] Goldberg, D., Reducibility of induced representations for Sp(2n) and SO(n). Amer. J. Math. 116 (1994), 1101–1151. Google Scholar

[Go2] [Go2] Goldberg, D., Reducibility for non-connected p-adic groups with G0of prime index. Canad. J. Math. 43 (1995), 344–363. Google Scholar

[I-M] [I-M] Iwahori, N. and Matsumoto, H., On some Bruhat decomposition and the structure of the Hecke rings of p-adic Chevalley groups. Inst. Hautes É tudes Sci. Publ. Math. 25 (1965), 5–48. Google Scholar

[J1] [J1] Jantzen, C., Degenerate principal series for symplectic and odd-orthogonal groups. Mem. Amer. Math. Soc. 590(1996). Google Scholar

[J2] [J2] Jantzen, C., Reducibility of certain representations for symplectic and odd-orthogonal groups. Compositio Math. 104 (1996), 55–63. Google Scholar

[J3] [J3] Jantzen, C., On supports of induced representations for symplectic and odd-orthogonal groups. Amer. J. Math. 119 (1997), 1213–1262. Google Scholar

[J4] [J4] Jantzen, C., On square-integrable representations of classical p-adic groups. Canad. J. Math. 52 (2000), 539–581. Google Scholar

[J5] [J5] Jantzen, C., On square-integrable representations of classical p-adic groups. Represent. Theory 4 (2000), 127–180. Google Scholar

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

[Kt] [Kt] Kato, S., Duality for representations of a Hecke algebra. Proc. Amer.Math. Soc. 119 (1993), 941–946. Google Scholar

[Kw] [Kw] Kawanaka, N., Fourier transforms of nilpotently supported invariant functions on a simple Lie algebra over a finite field. Invent.Math. 69 (1982), 411–435. Google Scholar

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

[Mu] [Mu] Muić, G., The unitary dual of p-adic G2. Duke Math. J. 90 (1997), 465–493. Google Scholar

[S-S] [S-S] 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

[Si1] [Si1] Silberger, A., The Langlands quotient theorem for p-adic groups. Math. Ann. 236 (1978), 95–104. Google Scholar

[Si2] [Si2] Silberger, A., Introduction to Harmonic Analysis on Reductive p-adic Groups. Princeton University Press, Princeton, NJ, 1979. Google Scholar

[T1] [T1] Tadić, M., Structure arising from induction and Jacquet modules of representations of classical p-adic groups. J. Algebra 177 (1995), 1–33. Google Scholar

[T2] [T2] Tadić, M., On reducibility of parabolic induction. Israel J. Math. 107 (1998), 29–91. Google Scholar

[T3] [T3] Tadić, M., On regular square integrable representations of p-adic groups. Amer. J. Math. 120 (1998), 159–210. Google Scholar

[Ze] [Ze] Zelevinsky, A., 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