Invariant distributions on \(p\)-adic analytic groups. (English) Zbl 1133.11066

Kohlhaase, Jan

Duke Math. J. 137, No. 1, 19-62 (2007).

This article is based on the author’s doctoral thesis [J. Kohlhaase, Invariant distributions on \(p\)-adic analytic groups, Münster: Univ. Münster, Fachbereich Mathematik und Informatik, Mathematisch-Naturwissenschaftliche Fakultät (Dissertation) (2005; Zbl 1088.11089)]. The work concerns \(p\)-adic distributions on a \(p\)-adic group \(G\). More precisely, \(G\) is the group of \(L\)-rational points of a split reductive group over \(L\), where \(L\) is a finite extension of \(\mathbb Q_p\). For a spherically complete extension \(K\) of \(L\), the author studies the algebra \(D(G,K)\) of locally analytic distributions on \(G\) with values in \(K\).
From the summary: “We derive several explicit descriptions of the center of the algebra \(D(G,K)\) of locally analytic distributions on \(G\) with values in \(K\). The main result is a generalization of an isomorphism of Harish-Chandra which connects the center of \(D(G,K)\) with the algebra of Weyl-invariant, centrally supported distributions on a maximal torus of \(G\). This isomorphism is supposed to play a role in the theory of locally analytic representations of \(G\) as studied by P. Schneider and J. Teitelbaum.”

Reviewer: Dubravka Ban (Carbondale)

MSC:

11S80	Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.)
22E50	Representations of Lie and linear algebraic groups over local fields

Keywords:

\(p\)-adic distributions; \(p\)-adic groups; locally analytic distributions

Citations:

Zbl 1088.11089

References:

[1]	M. Andler, A. Dvorsky, and S. Sahi, Kontsevich quantization and invariant distributions on Lie groups , Ann. Sci. École Norm. Sup. (4) 35 (2002), 371–390. · Zbl 1009.22020
[2]	L. Berger and C. Breuil, Towards a \(p\)-adic Langlands programme ,
[3]	S. Bosch, U. GüNtzer, and R. Remmert, Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry , Grundlehren Math. Wiss. 261 , Springer, Berlin, 1984. · Zbl 0539.14017
[4]	N. Bourbaki, Éléments de mathématique, fasc. 36: Variétés differentielles et analytiques: Fascicule de résultats (Paragraphes 8–15) , Actualités Sci. Indust. 1347 , Hermann, Paris, 1971. · Zbl 0217.20401
[5]	-, Éléments de mathématique, fasc. 37: Groupes et algèbres de Lie, chapitre 2: Algèbres de Lie libres; chapitre 3: Groupes de Lie , Actualités Sci. Indust. 1349 , Hermann, Paris, 1972. · Zbl 0244.22007
[6]	-, Elements of Mathematics: Commutative Algebra , Hermann, Paris, 1972. · Zbl 0279.13001
[7]	-, Topological Vector Spaces, chapters 1–5 , Springer, Berlin, 1987. · Zbl 0643.54012
[8]	J. Dixmier, Algèbres enveloppantes , Cahiers Sci. 37 , Gauthiers-Villars, Paris, 1974. · Zbl 0308.17007
[9]	J. D. Dixon, M. P. F. Du Sautoy, A. Mann, and D. Segal, Analytic Pro-\(p\) Groups , 2nd ed., Cambridge Stud. Adv. Math. 61 , Cambridge Univ. Press, Cambridge, 1999. · Zbl 0934.20001
[10]	M. Duflo, Caractères des groupes et algèbres de Lie résolubles , Ann. Sci. École Norm. Sup. (4) 3 (1970), 23–74. · Zbl 0223.22016
[11]	-, Opérateurs différentiels bi-invariants sur un groupe de Lie , Ann. Sci. École Norm. Sup. (4) 10 (1977), 265–288. · Zbl 0353.22009
[12]	M. Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms , Invent. Math. 164 (2006), 1–84. · Zbl 1090.22008
[13]	-, Locally analytic vectors in representations of locally \(p\)-adic analytic groups , to appear in Mem. Amer. Math. Soc.,
[14]	C. T. FéAux De Lacroix, Einige Resultate über die topologischen Darstellungen \(p\)-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem \(p\)-adischen Körper , Schriftenreihe Math. Inst. Univ. Münster 3. Ser. 23 , Univ. Münster, Münster 1999. · Zbl 0963.22009
[15]	J. Fresnel and M. Van Der Put, Rigid Analytic Geometry and Its Applications , Progr. Math. 218 , Birkhäuser, Boston, 2004. · Zbl 1096.14014
[16]	H. Frommer, The locally analytic principal series of split reductive groups , preprint, 2003.
[17]	A. Grothendieck, Produits tensoriels topologiques et espaces nucléaires , Mem. Amer. Math. Soc. 1955 , no. 16. · Zbl 0123.30301
[18]	A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, III: Étude cohomologique des faisceaux cohérents, I , Inst. Hautes Études Sci. Publ. Math. 11 (1961).
[19]	R. Kiehl, Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie , Invent. Math. 2 (1967), 256–273. · Zbl 0202.20201
[20]	U. KöPf, Über eigentliche Familien algebraischer Varietäten über affinoiden Räumen , Schr. Math. Inst. Univ. Münster (2) 7 (1974). · Zbl 0275.14006
[21]	A. M. Robert, A Course in \(p\)-Adic Analysis , Grad. Texts in Math. 198 , Springer, New York, 2000.
[22]	P. Schneider, Nonarchimedean Functional Analysis , Springer Monogr. Math., Springer, Berlin, 2002. · Zbl 0998.46044
[23]	-, \(p\)-Adische Analysis , lecture notes, Universität Münster, Münster, 2001.
[24]	P. Schneider and J. Teitelbaum, \(p\)-adic Fourier theory , Doc. Math. 6 (2001), 447–481. · Zbl 1028.11069
[25]	-, Locally analytic distributions and \(p\)-adic representation theory , with applications to GL\(_2\), J. Amer. Math. Soc. 15 (2002), 443–468. JSTOR: · Zbl 1028.11071
[26]	-, “\(p\)-adic boundary values” in Cohomologies \(p\)-adiques et applications arithmétiques, I , Astérisque 278 , Soc. Math. France, Montrouge, 2002, 51–125.; Correction in Cohomologies \(p\)-adiques et applications arithmétiques, III , Astérisque 295 , Soc. Math. France, Montrouge, 2004, 291–299. ; Mathematical Reviews (MathSciNet):
[27]	-, Algebras of \(p\)-adic distributions and admissible representations , Invent. Math. 153 (2003), 145–196. · Zbl 1028.11070
[28]	-, Duality for admissible locally analytic representations , Represent. Theory 9 (2005), 297–326. · Zbl 1146.22301
[29]	K.-Y. C. Sit, On bounded elements of linear algebraic groups , Trans. Amer. Math. Soc. 209 (1975), 185–198. · Zbl 0273.22005