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Journal of Lie Theory 30 (2020), No. 1, 085--144
Copyright Heldermann Verlag 2020

Smooth Duality and Co-Contra 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

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 p-adic 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 semi-infinite homological algebra [see: L. Positselski: Homological Algebra of Semimodules and Semicontramodules: Semi-infinite Homological Algebra of Associative Algebraic Structures, Monografie Matematyczne IMPAN 70, Birkhäuser, Basel (2010)].

Keywords: Locally profinite groups, p-adic Lie groups, modular representation theory, coalgebras and semialgebras, comodules and contramodules, admissible representations, derived categories, involutive triangulated duality.

MSC: 22E50, 22D35, 20E18, 16T15, 18E30.

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