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\(U({\mathfrak g})\)-finite locally analytic representations. (English) Zbl 1028.17007
Summary: We continue our algebraic approach to the study of locally analytic representations of a \(p\)-adic Lie group \(G\) in vector spaces over a non-Archimedean complete field \(K\). We characterize the smooth representations of Langlands theory which are contained in the new category. More generally, we completely determine the structure of the representations on which the universal enveloping algebra \(U(\mathfrak g)\) of the Lie algebra \(\mathfrak g\) of \(G\) acts through a finite dimensional quotient. They are direct sums of tensor products of smooth and rational \(G\)-representations. Finally we analyze the reducible members of the principal series of the group \(G= \text{SL}_2(\mathbb Q_p)\) in terms of such tensor products.

17B15 Representations of Lie algebras and Lie superalgebras, analytic theory
22E50 Representations of Lie and linear algebraic groups over local fields
17B35 Universal enveloping (super)algebras
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[1] 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. · Zbl 0622.46001
[2] Charles W. Curtis and Irving Reiner, Representation theory of finite groups and associative algebras, Pure and Applied Mathematics, Vol. XI, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1962. · Zbl 0131.25601
[3] 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. · Zbl 0203.23401
[4] Jacques Dixmier, Enveloping algebras, Graduate Studies in Mathematics, vol. 11, American Mathematical Society, Providence, RI, 1996. Revised reprint of the 1977 translation. · Zbl 0867.17001
[5] Christian Tobias Féaux de Lacroix, Einige Resultate über die topologischen Darstellungen \?-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem \?-adischen Körper, Schriftenreihe des Mathematischen Instituts der Universität Münster. 3. Serie, Heft 23, Schriftenreihe Math. Inst. Univ. Münster 3. Ser., vol. 23, Univ. Münster, Math. Inst., Münster, 1999, pp. x+111 (German). · Zbl 0963.22009
[6] I. M. Gel\(^{\prime}\)fand, M. I. Graev, and I. I. Pyatetskii-Shapiro, Representation theory and automorphic functions, Generalized Functions, vol. 6, Academic Press, Inc., Boston, MA, 1990. Translated from the Russian by K. A. Hirsch; Reprint of the 1969 edition.
[7] Harish-Chandra, Harmonic analysis on reductive \?-adic groups, Lecture Notes in Mathematics, Vol. 162, Springer-Verlag, Berlin-New York, 1970. Notes by G. van Dijk. · Zbl 0202.41101
[8] Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. · Zbl 0654.20039
[9] Yasuo Morita, Analytic representations of \?\?\(_{2}\) over a \?-adic number field. III, Automorphic forms and number theory (Sendai, 1983) Adv. Stud. Pure Math., vol. 7, North-Holland, Amsterdam, 1985, pp. 185 – 222. · Zbl 0621.22016
[10] Schneider, P., Teitelbaum, J.: Locally analytic distributions and \(p\)-adic representation theory, with applications to \(GL_{2}\). Preprint, 1999. · Zbl 1028.11071
[11] Marie-France Vignéras, Représentations \?-modulaires d’un groupe réductif \?-adique avec \?\ne \?, Progress in Mathematics, vol. 137, Birkhäuser Boston, Inc., Boston, MA, 1996 (French, with English summary). · Zbl 0859.22001
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