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Locally analytic distributions and $p\/$-adic representation theory, with applications to $GL_{2}$

Authors: Peter Schneider and Jeremy Teitelbaum
Journal: J. Amer. Math. Soc. 15 (2002), 443-468
MSC (2000): Primary 11S80, 22E50
Published electronically: October 18, 2001
MathSciNet review: 1887640
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Abstract: In this paper we study continuous representations of locally $L$-analytic groups $G$ in locally convex $K$-vector spaces, where $L$ is a finite extension of $\mathbb{Q}_p$ and $K$ is a spherically complete nonarchimedean extension field of $L$. The class of such representations includes both the smooth representations of Langlands theory and the finite dimensional algebraic representations of $G$, along with interesting new objects such as the action of $G$ on global sections of equivariant vector bundles on $p$-adic symmetric spaces. We introduce a restricted category of such representations that we call ``strongly admissible'' and we show that, when $G$ is compact, our category is anti-equivalent to a subcategory of the category of modules over the locally analytic distribution algebra of $G$. As an application we prove the topological irreducibility of generic members of the $p$-adic principal series for $GL_2(\mathbb{Q}_p)$. Our hope is that our definition of strongly admissible representation may be used as a foundation for a general theory of continuous $K$-valued representations of locally $L$-analytic groups.

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  • [Am1] Yvette Amice, Interpolation 𝑝-adique, Bull. Soc. Math. France 92 (1964), 117–180 (French). MR 188199
  • [Am2] Yvette Amice, Duals, Proceedings of the Conference on 𝑝-adic Analysis (Nijmegen, 1978), Report, vol. 7806, Katholieke Univ., Nijmegen, 1978, pp. 1–15. MR 522117
  • [B-GAL] N. Bourbaki, Éléments de mathématique. Fasc. XXVI. Groupes et algèbres de Lie. Chapitre I: Algèbres de Lie, Seconde édition. Actualités Scientifiques et Industrielles, No. 1285, Hermann, Paris, 1971 (French). MR 0271276
  • [B-TVS] N. Bourbaki, Topological vector spaces. Chapters 1–5, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1987. Translated from the French by H. G. Eggleston and S. Madan. MR 910295
  • [B-VAR] N. Bourbaki, Éléments de mathématique. Fasc. XXXIII. Variétés différentielles et analytiques. Fascicule de résultats (Paragraphes 1 à 7), Actualités Scientifiques et Industrielles, No. 1333, Hermann, Paris, 1967 (French). MR 0219078
  • [GKPS] N. De Grande-De Kimpe, J. Kakol, C. Perez-Garcia, and W. H. Schikhof, 𝑝-adic locally convex inductive limits, 𝑝-adic functional analysis (Nijmegen, 1996) Lecture Notes in Pure and Appl. Math., vol. 192, Dekker, New York, 1997, pp. 159–222. MR 1459211
  • [DG] Michel Demazure and Pierre Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris; North-Holland Publishing Co., Amsterdam, 1970 (French). Avec un appendice Corps de classes local par Michiel Hazewinkel. MR 0302656
  • [Fe1] Féaux de Lacroix, C. T., $p$-adische Distributionen, Diplomarbeit, Köln 1992.
  • [Fe2] Féaux de Lacroix, C. T., Einige Resultate über die topologischen Darstellungen $p$-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem $p$-adischen Körper, Thesis, Köln 1997, Schriftenreihe Math. Inst. Univ. Münster, 3. Serie, Heft 23, pp. 1-111 (1999).
  • [Gru] Laurent Gruson, Théorie de Fredholm 𝑝-adique, Bull. Soc. Math. France 94 (1966), 67–95 (French). MR 226381
  • [Hel] Helmer, O., The elementary divisor theorem for certain rings without chain conditions, Bull. AMS 49, 225-236 (1943). MR 4:185d
  • [Jan] Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
  • [Kom] Hikosaburo Komatsu, Projective and injective limits of weakly compact sequences of locally convex spaces, J. Math. Soc. Japan 19 (1967), 366–383. MR 217557,
  • [Laz] Michel Lazard, Les zéros des fonctions analytiques d’une variable sur un corps valué complet, Inst. Hautes Études Sci. Publ. Math. 14 (1962), 47–75 (French). MR 152519
  • [Mor] Yasuo Morita, Analytic representations of 𝑆𝐿₂ over a 𝔭-adic number field. III, Automorphic forms and number theory (Sendai, 1983) Adv. Stud. Pure Math., vol. 7, North-Holland, Amsterdam, 1985, pp. 185–222. MR 876106,
  • [NFA] Schneider, P., Nonarchimedean Functional Analysis, Berlin-Heidelberg-New York: Springer 2001.
  • [Sch] Schneider, P., $p\/$-adic representation theory, The 1999 Britton Lectures at McMaster University. Available at schneider.
  • [ST] Schneider, P., Teitelbaum, J., $p\/$-adic boundary values, To appear in Astérisque.
  • [Ti1] J. van Tiel, Espaces localement 𝐾-convexes. I, Nederl. Akad. Wetensch. Proc. Ser. A 68 = Indag. Math. 27 (1965), 249–258 (French). MR 0179593
  • [Ti2] J. van Tiel, Ensembles pseudo-polaires dans les espaces localement 𝐾-convexes, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math. 28 (1966), 369–373 (French). MR 0198198

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Additional Information

Peter Schneider
Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany

Jeremy Teitelbaum
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607

Received by editor(s): December 16, 1999
Received by editor(s) in revised form: May 16, 2001
Published electronically: October 18, 2001
Article copyright: © Copyright 2001 American Mathematical Society