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Dimensions of some locally analytic representations


Authors: Tobias Schmidt and Matthias Strauch
Journal: Represent. Theory 20 (2016), 14-38
MSC (2010): Primary 11E95, 22E50; Secondary 11S80, 16S30
DOI: https://doi.org/10.1090/ert/475
Published electronically: February 2, 2016
MathSciNet review: 3455080
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Abstract: Let $ G$ be the group of points of a split reductive group over a finite extension of $ \mathbb{Q}_p$. In this paper, we compute the dimensions of certain classes of locally analytic $ G$-representations. This includes principal series representations and certain representations coming from homogeneous line bundles on $ p$-adic symmetric spaces. As an application, we compute the dimensions of the unitary $ {\rm GL}_2(\mathbb{Q}_p)$-representations appearing in Colmez' $ p$-adic local Langlands correspondence.


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  • [1] K. Ardakov, $ \hat {\mathcal D}$-modules on rigid analytic spaces, Proc. International Congress of Mathematicians 2014 (to appear).
  • [2] K. Ardakov and K. A. Brown, Ring-theoretic properties of Iwasawa algebras: a survey, Doc. Math. Extra Vol. (2006), 7-33. MR 2290583 (2007k:11185)
  • [3] Konstantin Ardakov and Simon Wadsley, On irreducible representations of compact $ p$-adic analytic groups, Ann. of Math. (2) 178 (2013), no. 2, 453-557. MR 3071505, https://doi.org/10.4007/annals.2013.178.2.3
  • [4] Alexandre Beilinson and Joseph Bernstein, Localisation de $ g$-modules, C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), no. 1, 15-18 (French, with English summary). MR 610137 (82k:14015)
  • [5] Pierre Berthelot, $ {\mathcal {D}}$-modules arithmétiques. I. Opérateurs différentiels de niveau fini, Ann. Sci. École Norm. Sup. (4) 29 (1996), no. 2, 185-272 (French, with English summary). MR 1373933 (97b:14019)
  • [6] Christophe Breuil, Sur quelques représentations modulaires et $ p$-adiques de $ {\rm GL}_2(\mathbf {Q}_p)$. II, J. Inst. Math. Jussieu 2 (2003), no. 1, 23-58 (French, with French summary). MR 1955206 (2005d:11079), https://doi.org/10.1017/S1474748003000021
  • [7] J.-L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math. 64 (1981), no. 3, 387-410. MR 632980 (83e:22020), https://doi.org/10.1007/BF01389272
  • [8] David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060 (94j:17001)
  • [9] Pierre Colmez, Gabriel Dospinescu, and Vytautas Paškūnas, The $ p$-adic local Langlands correspondence for $ {\rm GL}_2(\mathbb{Q}_p)$, Camb. J. Math. 2 (2014), no. 1, 1-47. MR 3272011, https://doi.org/10.4310/CJM.2014.v2.n1.a1
  • [10] P. Colmez, La série principale unitaire de $ {\rm GL}\sb 2(\mathbf Q\sb p)$: vecteurs localement analytiques, Automorphic Forms and Galois Representations, Vol. 1, London Math. Soc. Lect. Note Series 415 (2014), 286-358.
  • [11] Pierre Colmez, La série principale unitaire de $ {\rm GL}_2(\mathbf {Q}_p)$, Astérisque 330 (2010), 213-262 (French, with English and French summaries). MR 2642407 (2011g:22026)
  • [12] P. Colmez, Représentations de $ {\rm GL}\sb 2({\mathbf Q}\sb p)$ et $ (\phi ,\Gamma )$-modules, Astérisque (2010), no. 330, 281-509.
  • [13] Gabriel Dospinescu, Actions infinitésimales dans la correspondance de Langlands locale $ p$-adique, Math. Ann. 354 (2012), no. 2, 627-657 (French, with English summary). MR 2965255, https://doi.org/10.1007/s00208-011-0736-2
  • [14] M. Emerton, Locally analytic vectors in representations of locally $ p$-adic analytic groups, Preprint. To appear in: Memoirs of the AMS.
  • [15] A. Ya. Helemskii, Banach and locally convex algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. Translated from the Russian by A. West. MR 1231796 (94f:46001)
  • [16] James E. Humphreys, Representations of semisimple Lie algebras in the BGG category $ \mathcal {O}$, Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008. MR 2428237 (2009f:17013)
  • [17] Jens Carsten Jantzen, Einhüllende Algebren halbeinfacher Lie-Algebren, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 3, Springer-Verlag, Berlin, 1983 (German). MR 721170 (86c:17011)
  • [18] Jens Carsten Jantzen, Representations of Lie algebras in prime characteristic, Representation theories and algebraic geometry (Montreal, PQ, 1997), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 514, Kluwer Acad. Publ., Dordrecht, 1998, pp. 185-235. Notes by Iain Gordon. MR 1649627 (99h:17026)
  • [19] A. Joseph, Goldie rank in the enveloping algebra of a semisimple Lie algebra. I, II, J. Algebra 65 (1980), no. 2, 269-283, 284-306. MR 585721 (82f:17009), https://doi.org/10.1016/0021-8693(80)90217-3
  • [20] David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979), no. 2, 165-184. MR 560412 (81j:20066), https://doi.org/10.1007/BF01390031
  • [21] J. Lepowsky, Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism, J. Algebra 49 (1977), no. 2, 470-495. MR 0463360 (57 #3312)
  • [22] Thierry Levasseur, Some properties of noncommutative regular graded rings, Glasgow Math. J. 34 (1992), no. 3, 277-300. MR 1181768 (93k:16045), https://doi.org/10.1017/S0017089500008843
  • [23] Huishi Li and Freddy van Oystaeyen, Zariskian filtrations, $ K$-Monographs in Mathematics, vol. 2, Kluwer Academic Publishers, Dordrecht, 1996. MR 1420862 (97m:16083)
  • [24] Huishi Li, Lifting Ore sets of Noetherian filtered rings and applications, J. Algebra 179 (1996), no. 3, 686-703. MR 1371738 (97a:16088), https://doi.org/10.1006/jabr.1996.0031
  • [25] Ruochuan Liu, Bingyong Xie, and Yuancao Zhang, Locally analytic vectors of unitary principal series of $ {\rm GL}_2(\mathbb{Q}_p)$, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), no. 1, 167-190 (English, with English and French summaries). MR 2961790
  • [26] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572 (89j:16023)
  • [27] Sascha Orlik, Equivariant vector bundles on Drinfeld's upper half space, Invent. Math. 172 (2008), no. 3, 585-656. MR 2393081 (2009c:22019), https://doi.org/10.1007/s00222-008-0112-3
  • [28] Sascha Orlik and Matthias Strauch, On Jordan-Hölder series of some locally analytic representations, J. Amer. Math. Soc. 28 (2015), no. 1, 99-157. MR 3264764, https://doi.org/10.1090/S0894-0347-2014-00803-1
  • [29] Vytautas Paškūnas, Admissible unitary completions of locally $ \mathbb{Q}_p$-rational representations of $ {\rm GL}_2(F)$, Represent. Theory 14 (2010), 324-354. MR 2608966 (2011i:22017), https://doi.org/10.1090/S1088-4165-10-00373-0
  • [30] Vytautas Paškūnas, The image of Colmez's Montreal functor, Publ. Math. Inst. Hautes Études Sci. 118 (2013), 1-191. MR 3150248, https://doi.org/10.1007/s10240-013-0049-y
  • [31] George S. Rinehart, Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195-222. MR 0154906 (27 #4850)
  • [32] Tobias Schmidt, Auslander regularity of $ p$-adic distribution algebras, Represent. Theory 12 (2008), 37-57. MR 2375595 (2009b:22018), https://doi.org/10.1090/S1088-4165-08-00323-3
  • [33] Tobias Schmidt, Analytic vectors in continuous $ p$-adic representations, Compos. Math. 145 (2009), no. 1, 247-270. MR 2480502 (2009k:22025), https://doi.org/10.1112/S0010437X08003825
  • [34] Tobias Schmidt, Stable flatness of nonarchimedean hyperenveloping algebras, J. Algebra 323 (2010), no. 3, 757-765. MR 2574862 (2011a:17023), https://doi.org/10.1016/j.jalgebra.2009.11.018
  • [35] Tobias Schmidt, Verma modules over $ p$-adic Arens-Michael envelopes of reductive Lie algebras, J. Algebra 390 (2013), 160-180. MR 3072116, https://doi.org/10.1016/j.jalgebra.2013.04.038
  • [36] Tobias Schmidt, On locally analytic Beilinson-Bernstein localization and the canonical dimension, Math. Z. 275 (2013), no. 3-4, 793-833. MR 3127038, https://doi.org/10.1007/s00209-013-1161-x
  • [37] Peter Schneider, Nonarchimedean functional analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1869547 (2003a:46106)
  • [38] P. Schneider, Continuous representation theory of $ p$-adic Lie groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1261-1282.
  • [39] P. Schneider and J. Teitelbaum, Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359-380. MR 1900706 (2003c:22026), https://doi.org/10.1007/BF02784538
  • [40] Peter Schneider and Jeremy Teitelbaum, Locally analytic distributions and $ p$-adic representation theory, with applications to $ {\rm GL}_2$, J. Amer. Math. Soc. 15 (2002), no. 2, 443-468 (electronic). MR 1887640 (2003b:11132), https://doi.org/10.1090/S0894-0347-01-00377-0
  • [41] Peter Schneider and Jeremy Teitelbaum, Algebras of $ p$-adic distributions and admissible representations, Invent. Math. 153 (2003), no. 1, 145-196. MR 1990669 (2004g:22015), https://doi.org/10.1007/s00222-002-0284-1
  • [42] Peter Schneider and Jeremy Teitelbaum, Duality for admissible locally analytic representations, Represent. Theory 9 (2005), 297-326 (electronic). MR 2133762 (2006a:22016), https://doi.org/10.1090/S1088-4165-05-00277-3
  • [43] Otmar Venjakob, On the structure theory of the Iwasawa algebra of a $ p$-adic Lie group, J. Eur. Math. Soc. (JEMS) 4 (2002), no. 3, 271-311. MR 1924402 (2004h:16029), https://doi.org/10.1007/s100970100038

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Additional Information

Tobias Schmidt
Affiliation: Institute de Recherche Mathématique de Rennes, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France
Email: Tobias.Schmidt@univ-rennes1.fr

Matthias Strauch
Affiliation: Department of Mathematics, Indiana University, Rawles Hall, Bloomington, Indiana 47405
Email: mstrauch@indiana.edu

DOI: https://doi.org/10.1090/ert/475
Received by editor(s): December 18, 2014
Received by editor(s) in revised form: November 17, 2015, and December 11, 2015
Published electronically: February 2, 2016
Additional Notes: The first author would like to acknowledge support of the Heisenberg programme of Deutsche Forschungsgemeinschaft
The second author would like to acknowledge the support of the National Science Foundation (award DMS-1202303).
Article copyright: © Copyright 2016 American Mathematical Society

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