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Analytic vectors in continuous p-adic representations

Part of: Discontinuous groups and automorphic forms Algebraic number theory: local and $p$-adic fields Lie groups

Published online by Cambridge University Press:  01 January 2009

Tobias Schmidt
Tobias Schmidt*
Affiliation:
Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstraße 62, D-48149 Münster, Germany (email: [email protected])
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Abstract

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Given a compact p-adic Lie group G over a finite unramified extension L/ℚp let GL/ℚp be the product over all Galois conjugates of G. We construct an exact and faithful functor from admissible G-Banach space representations to admissible locally L-analytic GL/ℚp-representations that coincides with passage to analytic vectors in the case L=ℚp. On the other hand, we study the functor ‘passage to analytic vectors’ and its derived functors over general basefields. As an application we compute the higher analytic vectors in certain locally analytic induced representations.

Keywords

p-adic Lie groupsrepresentation theoryanalytic vector

MSC classification

Secondary: 22E50: Representations of Lie and linear algebraic groups over local fields 11S99: None of the above, but in this section 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 1 , January 2009 , pp. 247 - 270
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Bourbaki, N., Variétés différentielles et analytiques, in Fascicule de résultats (Hermann, Paris, 1967).Google Scholar
[2]Bourbaki, N., Topological vector spaces (Springer, Berlin, 2003), chs 1–5.CrossRefGoogle Scholar
[3]Borel, A. and Wallach, N., Continuous cohomology, discrete subgroups, and representations of reductive groups (Princeton University Press, Princeton, NJ, 1980).Google Scholar
[4]Dixmier, J., Cohomologie des algèbres de Lie nilpotent, Acta. Sci. Math. 16 (1995), 246–250.Google Scholar
[5]Dixon, J. D., du Sautoy, M. P. F., Mann, A. and Segal, D., Analytic pro-p groups (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
[6]Emerton, M., Locally analytic vectors in representations of non-archimedean locally p-adic analytic groups, Mem. Amer. Math. Soc. to appear.Google Scholar
[7]Emerton, M., Locally analytic representation theory of p-adic reductive groups: a summary of some recent developments, in L-functions and Galois representations, London Mathematical Society Lecture Note Series, vol. 320, eds D. Burns, K. Buzzard and J. Nekovar (2007), 407–437.CrossRefGoogle Scholar
[8]Frommer, H., The locally analytic principal series of split reductive groups, Münster, SFB-preprint 265 (2003). Available at: http://wwwmath1.uni-muenster.de/sfb/about/publ/.Google Scholar
[9]Féaux de Lacroix, C. T., Einige Resultate über die topologischen Darstellungen p-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem p-adischen Körper, Schrift. Math. Inst. Univ. Münster, 3 Ser. 23 (1999), 1–111.Google Scholar
[10]Hochschild, G. and Serre, J.-P., Cohomologie of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603.CrossRefGoogle Scholar
[11]Kohlhaase, J., Invariant distributions on p-adic analytic groups, Duke Math. J. 137 (2007), 19–62.CrossRefGoogle Scholar
[12]Kohlhaase, J., The cohomology of locally analytic representations, Münster, SFB-preprint 491 (2007). Available at: http://wwwmath1.uni-muenster.de/sfb/about/publ/.Google Scholar
[13]Li, H. and van Oystaeyen, F., Zariskian filtrations (Kluwer, Dordrecht, 1996).Google Scholar
[14]Schneider, P., Nonarchimedean functional analysis (Springer, Berlin, 2002).CrossRefGoogle Scholar
[15]Orlik, S. and Strauch, M., On the irreducibiliy of locally analytic principal series representations, Represent. Theory, to appear.Google Scholar
[16]Pirkovski, A. Y., Stably flat completions of universal enveloping algebras, Diss. Math. 441 (2006), 1–60.Google Scholar
[17]Schneider, P., Continuous representation theory of p-adic Lie groups, in Proc. int. conf. Mathematics, Madrid, 2006, vol. II. 2006.Google Scholar
[18]Schmidt, T., Auslander Regularity of p-adic Distribution Algebras, Represent. Theory 12 (2008), 35–57.CrossRefGoogle Scholar
[19]Schneider, P. and Teitelbaum, J., U(𝔤)-finite locally analytic representations, Represent. Theory 5 (2001), 111–128.CrossRefGoogle Scholar
[20]Schneider, P. and Teitelbaum, J., p-adic Fourier theory, Documenta Math. 6 (2001), 447–481.CrossRefGoogle Scholar
[21]Schneider, P. and Teitelbaum, J., Locally analytic distributions and p-adic representation theory, with applications to GL2, J. Amer. Math. Soc. 15 (2002), 443–468.CrossRefGoogle Scholar
[22]Schneider, P. and Teitelbaum, J., Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359–380.CrossRefGoogle Scholar
[23]Schneider, P. and Teitelbaum, J., Algebras of p-adic distributions and admissible representations, Inv. Math. 153 (2003), 145–196.CrossRefGoogle Scholar
[24]Schneider, P. and Teitelbaum, J., Duality for admissible locally analytic representations, Represent. Theory 9 (2005), 297–326.CrossRefGoogle Scholar
[25]Teitelbaum, J., Admissible analytic representations. An introduction and three questions. Talk at Harvard Eigensemester 2006. Available at: http://www2.math.uic.edu/ jeremy/harvard.Google Scholar