On Jordan-Hölder series of some locally analytic representations

Orlik, Sascha; Strauch, Matthias

doi:10.1090/S0894-0347-2014-00803-1

On Jordan-Hölder series of some locally analytic representations
HTML articles powered by AMS MathViewer

by Sascha Orlik and Matthias Strauch

J. Amer. Math. Soc. 28 (2015), 99-157

DOI: https://doi.org/10.1090/S0894-0347-2014-00803-1

Published electronically: July 2, 2014

PDF | Request permission

Abstract:

Let $G$ be a split reductive $p$-adic group. This paper is about the Jordan-Hölder series of locally analytic $G$-representations which are induced from locally algebraic representations of a parabolic subgroup $P \subset G$. We construct for every representation $M$ of $\textrm {Lie}(G)$ in the BGG-category ${\mathcal O}$, which is equipped with an algebraic $P$-action, and for every smooth $P$-representation $V$, a locally analytic representation ${\mathcal F}^G_P(M,V)$ of $G$. This gives rise to a bi-functor to the category of locally analytic representations. We prove that it is exact and give a criterion for the topological irreducibility of ${\mathcal F}^G_P(M,V)$ in terms of $M$ and $V$.

References

H. H. Andersen and N. Lauritzen, Twisted Verma modules, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000) Progr. Math., vol. 210, Birkhäuser Boston, Boston, MA, 2003, pp. 1–26. MR 1985191
I. N. Bernšteĭn, I. M. Gel′fand, and S. I. Gel′fand, A certain category of ${\mathfrak {g}}$-modules, Funkcional. Anal. i Priložen. 10 (1976), no. 2, 1–8 (Russian). MR 0407097
A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, 2nd ed., Mathematical Surveys and Monographs, vol. 67, American Mathematical Society, Providence, RI, 2000. MR 1721403, DOI 10.1090/surv/067
Raoul Bott, Homogeneous vector bundles, Ann. of Math. (2) 66 (1957), 203–248. MR 89473, DOI 10.2307/1969996
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626
W. Casselman, On a $p$-adic vanishing theorem of Garland, Bull. Amer. Math. Soc. 80 (1974), 1001–1004. MR 354933, DOI 10.1090/S0002-9904-1974-13611-6
Ivan Dimitrov, Olivier Mathieu, and Ivan Penkov, On the structure of weight modules, Trans. Amer. Math. Soc. 352 (2000), no. 6, 2857–2869. MR 1624174, DOI 10.1090/S0002-9947-00-02390-4
J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic pro-$p$ groups, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 61, Cambridge University Press, Cambridge, 1999. MR 1720368, DOI 10.1017/CBO9780511470882
Emerton M., Jacquet modules of locally analytic representations of $p$-adic reductive groups II. The relation to parabolic induction. Preprint.
Emerton M., Locally analytic vectors in representations of locally $p$-adic analytic groups. To appear in Memoirs AMS., DOI 10.1090/memo/1175
Christian Tobias Féaux de Lacroix, Einige Resultate über die topologischen Darstellungen $p$-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem $p$-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). MR 1691735
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157, DOI 10.1007/978-1-4757-3849-0
James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, vol. 9, Springer-Verlag, New York-Berlin, 1978. Second printing, revised. MR 499562
James E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\scr {O}$, Graduate Studies in Mathematics, vol. 94, American Mathematical Society, Providence, RI, 2008. MR 2428237, DOI 10.1090/gsm/094
Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057
Owen T. R. Jones, An analogue of the BGG resolution for locally analytic principal series, J. Number Theory 131 (2011), no. 9, 1616–1640. MR 2802138, DOI 10.1016/j.jnt.2011.02.010
Anthony W. Knapp, Lie groups, Lie algebras, and cohomology, Mathematical Notes, vol. 34, Princeton University Press, Princeton, NJ, 1988. MR 938524
Jan Kohlhaase, Invariant distributions on $p$-adic analytic groups, Duke Math. J. 137 (2007), no. 1, 19–62. MR 2309143, DOI 10.1215/S0012-7094-07-13712-8
Jan Kohlhaase, The cohomology of locally analytic representations, J. Reine Angew. Math. 651 (2011), 187–240. MR 2774315, DOI 10.1515/CRELLE.2011.013
Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198, DOI 10.1007/978-1-4612-0105-2
J. Lepowsky, Generalized Verma modules, the Cartan-Helgason theorem, and the Harish-Chandra homomorphism, J. Algebra 49 (1977), no. 2, 470–495. MR 463360, DOI 10.1016/0021-8693(77)90253-8
Yasuo Morita, A $p$-adic theory of hyperfunctions. I, Publ. Res. Inst. Math. Sci. 17 (1981), no. 1, 1–24. MR 613933, DOI 10.2977/prims/1195186702
Yasuo Morita, Analytic representations of $\textrm {SL}_2$ over a ${\mathfrak {p}}$-adic number field. II, Automorphic forms of several variables (Katata, 1983) Progr. Math., vol. 46, Birkhäuser Boston, Boston, MA, 1984, pp. 282–297. MR 763019
Sascha Orlik, Equivariant vector bundles on Drinfeld’s upper half space, Invent. Math. 172 (2008), no. 3, 585–656. MR 2393081, DOI 10.1007/s00222-008-0112-3
Sascha Orlik and Matthias Strauch, On the irreducibility of locally analytic principal series representations, Represent. Theory 14 (2010), 713–746. MR 2738585, DOI 10.1090/S1088-4165-2010-00387-8
Pohlkamp K., Randwerte holomorpher Funktionen auf $p$-adischen symmetrischen Räumen (2004). Diplomarbeit, Universität Münster.
Tobias Schmidt, Auslander regularity of $p$-adic distribution algebras, Represent. Theory 12 (2008), 37–57. MR 2375595, DOI 10.1090/S1088-4165-08-00323-3
P. Schneider and U. Stuhler, The cohomology of $p$-adic symmetric spaces, Invent. Math. 105 (1991), no. 1, 47–122. MR 1109620, DOI 10.1007/BF01232257
P. Schneider, J. Teitelbaum, and Dipendra Prasad, $U({\mathfrak {g}})$-finite locally analytic representations, Represent. Theory 5 (2001), 111–128. With an appendix by Dipendra Prasad. MR 1835001, DOI 10.1090/S1088-4165-01-00109-1
Peter Schneider, Nonarchimedean functional analysis, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2002. MR 1869547, DOI 10.1007/978-3-662-04728-6
Peter Schneider, Continuous representation theory of $p$-adic Lie groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1261–1282. MR 2275644
Peter Schneider and Jeremy Teitelbaum, Locally analytic distributions and $p$-adic representation theory, with applications to $\textrm {GL}_2$, J. Amer. Math. Soc. 15 (2002), no. 2, 443–468. MR 1887640, DOI 10.1090/S0894-0347-01-00377-0
Peter Schneider and Jeremy Teitelbaum, $p$-adic boundary values, Astérisque 278 (2002), 51–125. Cohomologies $p$-adiques et applications arithmétiques, I. MR 1922824
Peter Schneider and Jeremy Teitelbaum, Algebras of $p$-adic distributions and admissible representations, Invent. Math. 153 (2003), no. 1, 145–196. MR 1990669, DOI 10.1007/s00222-002-0284-1
Peter Schneider and Jeremy Teitelbaum, Duality for admissible locally analytic representations, Represent. Theory 9 (2005), 297–326. MR 2133762, DOI 10.1090/S1088-4165-05-00277-3
Benjamin Schraen, Représentations localement analytiques de $\textrm {GL}_3(\Bbb Q_p)$, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), no. 1, 43–145 (French, with English and French summaries). MR 2760195, DOI 10.24033/asens.2140
Teitelbaum J., Admissible analytic representations. An introduction to three questions (2006). Talk at the Harvard Eigensemester.