Abstract
Let F be a P-adic field, let L be the completion of a maximal unramified extension of F, and let σ be the Frobenius automorphism of L over F. For any connected reductive group G over F one denotes by B(G) the set of σ-conjugacy classes in G(L) (elements x,y in G(L) are said to be σ-conjugate if there exists g in G(L) such that g-1κ σ(g)=y. One of the main results of this paper is a concrete description of the set B(G) (previously this was known only in the quasi-split case).
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References
[A] Adams, J. F.: Lectures on Lie Groups, Benjamin, New York, 1969.
[AV] Adams, J. and Vogan, D.: Lifting of characters and Harish-Chandra's method of descent, preprint.
[BS] Borel, A. and Serre, J-P.: Théorèmes de finitude en cohomologie galoisienne, Comm. Math. Helv.39 (1964) 111–164.
[B] Borovoi, M. V.: The algebraic fundamental group and abelian Galois cohomology of reductive algebraic groups, preprint.
[D] Deligne, P.: Catégories Tannakiennes, The Grothendieck Festschrift, vol. II, Birkhauser, Boston, 1990 pp. 111–195.
[K] Kottwitz, R.: Isocrystals with additional structure, Compositio Math.56 (1985) 201–220.
[K1] Kottwitz, R.: Shimura varieties and twisted orbital integrals, Math. Ann.269 (1984) 287–300.
[K2] Kottwitz, R.: Stable trace formula: elliptic singular terms, Math. Ann.275 (1986) 365–399.
[K3] Kottwitz, R.: Shimura Varieties and λ-adic Representations, Automorphic Forms, Shimura Varieties and L-functions, Part 1, Perspectives in Mathematics, vol. 10, Academic Press, San Diego, 1990 pp. 161–209.
[K4] Kottwitz, R.: Stable trace formula: cuspidal tempered terms, Duke Math. J.51 (1984) 611–650.
[K5] Kottwitz, R.: Points on some Shimura varieties over finite fields, J. Amer. Math. Soc.5 (1992) 373–444.
[Kn] Kneser, M.: Galoiskohomologie halbeinfacher algebraischerGruppen über p-adischenKörpern I, Math. Zeit.88 (1965) 40–47; II Math. Zeit.89 (1965) 250-272.
[KS] Kottwitz, R. and Shelstad, D.: Twisted endoscopy, preprint.
[L] Langlands, R. P.: On the zeta-functions of some simple Shimura varieties, Canad. J. Math.31 (1979) 1121–1216.
[LR] Langlands, R. P. and Rapoport, M.: Shimuravarietäten und Gerben, J. Reine Angew. Math.378 (1987) 113–220.
[R] Rapoport, M.: On the bad reduction of Shimura varieties, Automorphic Forms, Shimura Varieties and L-functions, Part2 Perspectives in Mathematics, vol. 10, Academic Press, San Diego, 1990 pp. 253–321.
[RR] Rapoport, M. and Richartz, M.: On the classification and specialization of F-isocrystals with additional structure, Compositio Math.. To appear.
[RZ] Rapoport, M. and Zink, T.: Period Spaces for p-Divisible Groups, Ann. of Math. Studies 141, Princeton University Press, 1996.
[Sa] Saavedra Rivano, N.: Catégories Tannakiennes, Lecture Notes in Mathematics 265, Springer-Verlag, 1972.
[S1] Serre, J-P.: Cohomologie Galoisienne, Lecture Notes in Mathematics 5, Springer-Verlag, 1965.
[S2] Serre, J-P.: Local Class Field Theory, Algebraic Number Theory, J. Cassels and A. Fröhlich (eds), Academic Press, 1967 pp. 128–161
[St] Steinberg, R.: Regular elements of semisimple algebraic groups, Inst. Hautes Études Sci. Publ. Math.25 (1965) 49–80.
[T] Tits, J.: Classification of Algebraic Semisimple Groups, Algebraic Groups and Discontinuous Subgroups, Proc. Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, RI, 1996 pp. 33ss–62.
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Kottwitz, R.E. Isocrystals with additional structure. II. Compositio Mathematica 109, 255–339 (1997). https://doi.org/10.1023/A:1000102604688
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DOI: https://doi.org/10.1023/A:1000102604688