Schneider-Teitelbaum duality for locally profinite groups

Document Type : Research Paper

Author

University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571 Japan

Abstract

We define monoidal structures on several categories of linear topological modules over the valuation ring of a local field, and study module theory with respect to the monoidal structures. We extend the notion of the Iwasawa algebra to a locally profinite group as a monoid with respect to one of the monoidal structure, which does not necessarily form a topological algebra. This is one of the main reasons why we need monoidal structures. We extend Schneider--Teitelbaum duality to duality applicable to a locally profinite group through the module theory over the generalised Iwasawa algebra, and give a criterion of the irreducibility of a unitary Banach representation.

Keywords


[1] Berkovich, V.G., Spectral Theory and Analytic Geometry over non-Archimedean Fields", Mathematical Surveys and Monographs 33, Amer. Math. Soc. 1990.
[2] Bosch, S., Guntzer, U., and Remmert, R., Non-Archimedean Analysis: A Systematic Approach to Rigid Analytic Geometry", Springer, 1984.
[3] Engelking, R., General Topology", Polish Scientific Publishers, 1977.
[4] Frank, D.L., A totally bounded, complete uniform space is compact, Proc. Amer. Math. Soc. 16 (1965), p. 514.
[5] Ingleton, A.W., The Hahn-Banach theorem for non-Archimedean valued fields, Math. Proc. Cambridge Philos. Soc. 48(1) (1952), 41-45.
[6] Mihara, T., Hahn-Banach theorem and duality theory on non-Archimedean locally convex spaces, J. Convex Anal. 24(2) (2017), 587-619.
[7] Mihara, T., Duality theory of p-adic Hopf algebras, Categ. General Algebraic Struct. Appl., to appear.
[8] Monna, A.F., Analyse Non-Archimedienne", Springer, 1970.
[9] Mostert, P.S., Local cross sections in locally compact groups, Proc. Amer. Math. Soc. 4(4) (1953), 645-649.
[10] Schikhof, W.H., A perfect duality between p-adic Banach spaces and compactoids, Indag. Math. (N.S.) 6(3) (1995), 325-339.
[11] Schneider, P., Non-Archimedean Functional Analysis", Springer, 2002.
[12] Demazure, M. and Grothendieck, A., Seminaire de Geometrie Algebrique du Bois Marie - 1962-64 - Schemas en groupes - SGA3 - Tome 1", Springer, 1970.
[13] Schneider, P. and Teitelbaum, J., Banach space representations and Iwasawa theory, Israel J. Math. 127(1) (2002), 359-380.