On the irreducibility of locally analytic principal series representations

Orlik, Sascha; Strauch, Matthias

doi:10.1090/S1088-4165-2010-00387-8

On the irreducibility of locally analytic principal series representations
HTML articles powered by AMS MathViewer

by Sascha Orlik and Matthias Strauch

Represent. Theory 14 (2010), 713-746

DOI: https://doi.org/10.1090/S1088-4165-2010-00387-8

Published electronically: December 1, 2010

PDF | Request permission

Abstract:

Let $\mathbf {G}$ be a $p$-adic connected reductive group with Lie algebra $\mathfrak {g}$. For a parabolic subgroup $\mathbf {P} \subset \mathbf {G}$ and a finite-dimensional locally analytic representation $V$ of a Levi subgroup of $\mathbf {P}$, we study the induced locally analytic $\mathbf {G}$-representation $W = \operatorname {Ind}_{\mathbf {P}}^{\mathbf {G}}(V)$. Our result is the following criterion concerning the topological irreducibility of $W$: If the Verma module $U(\mathfrak {g}) \otimes _{U(\mathfrak {p})} V’$ associated to the dual representation $V’$ is irreducible, then $W$ is topologically irreducible as well.

References

S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 261, Springer-Verlag, Berlin, 1984. A systematic approach to rigid analytic geometry. MR 746961, DOI 10.1007/978-3-642-52229-1
Christophe Breuil and Peter Schneider, First steps towards $p$-adic Langlands functoriality, J. Reine Angew. Math. 610 (2007), 149–180. MR 2359853, DOI 10.1515/CRELLE.2007.070
F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251 (French). MR 327923, DOI 10.1007/BF02715544
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
P. Cartier, Representations of $p$-adic groups: a survey, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 111–155. MR 546593
Jacques Dixmier, Enveloping algebras, North-Holland Mathematical Library, Vol. 14, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. Translated from the French. MR 0498740
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
M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, To appear in Memoirs of the AMS.
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
H. Frommer, The locally analytic principal series of split reductive groups, Preprintreihe SFB 478, Münster, Heft 265 (2003); available at http://3dsp.uni-muenster.de/wwwmath.uni-muenster.de/sfb/about/publ/frommer01.html.
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
J. Kohlhaase, The cohomology of locally analytic representations, preprint, Münster (2008); available at: http://wwwmath.uni-muenster.de/sfb/about/publ/kohlhaase.html, to appear in J. Reine Angew. Math. (Crelle).
Mark Kisin and Matthias Strauch, Locally analytic cuspidal representations for $\textrm {GL}_2$ and related groups, J. Inst. Math. Jussieu 5 (2006), no. 3, 373–421. MR 2241928, DOI 10.1017/S1474748005000228
Michel Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603 (French). MR 209286
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
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
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 and Ulrich Stuhler, Representation theory and sheaves on the Bruhat-Tits building, Inst. Hautes Études Sci. Publ. Math. 85 (1997), 97–191. MR 1471867, DOI 10.1007/BF02699536
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, 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
P. Schneider and J. Teitelbaum, $p$-adic Fourier theory, Doc. Math. 6 (2001), 447–481. MR 1871671
J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588