
Journal of Lie Theory 30 (2020), No. 1, 085144 Copyright Heldermann Verlag 2020 Smooth Duality and CoContra Correspondence Leonid Positselski Institute of Mathematics, Czech Academy of Sciences, 115~67 Prague 1, Czech Republic and: Laboratory of Algebra and Number Theory, Institute for Information Transmission Problems, Moscow 127051, Russia posic@mccme.ru The aim of this paper is to explain how to get a complex of smooth representations out of the dual vector space to a smooth representation of a padic Lie group, in natural characteristic. The construction does not depend on any finiteness/admissibility assumptions. Imposing such an assumption, one obtains an involutive duality on the derived category of complexes of smooth modules with admissible cohomology modules. The paper can serve as an introduction to the results about representations of locally profinite groups contained in the author's monograph on semiinfinite homological algebra [see: L. Positselski: Homological Algebra of Semimodules and Semicontramodules: Semiinfinite Homological Algebra of Associative Algebraic Structures, Monografie Matematyczne IMPAN 70, Birkhäuser, Basel (2010)]. Keywords: Locally profinite groups, padic Lie groups, modular representation theory, coalgebras and semialgebras, comodules and contramodules, admissible representations, derived categories, involutive triangulated duality. MSC: 22E50, 22D35, 20E18, 16T15, 18E30. [ Fulltextpdf (352 KB)] for subscribers only. 