Aubert, Anne-Marie Duality in the Grothendieck group of the category of finite length smooth representations of a reductive \(p\)-adic group. (Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif \(p\)- adique.) (French) Zbl 0827.22005 Trans. Am. Math. Soc. 347, No. 6, 2179-2189 (1995). Summary: We define an involution on the Grothendieck ring of the category of finite length smooth representations of a \(p\)-adic algebraic group, which is a direct analogue Curtis-Alvis duality for finite groups of Lie type. This involution commutes with taking the contragredient, with parabolic induction and, up a few twists, with truncation. It also preserves the irreducible representations up to sign. Cited in 10 ReviewsCited in 50 Documents MSC: 22E50 Representations of Lie and linear algebraic groups over local fields 18F30 Grothendieck groups (category-theoretic aspects) 20G05 Representation theory for linear algebraic groups Keywords:involution; Grothendieck ring; category of finite length smooth representations; \(p\)-adic algebraic group; parabolic induction; irreducible representations PDF BibTeX XML Cite \textit{A.-M. Aubert}, Trans. Am. Math. Soc. 347, No. 6, 2179--2189 (1995; Zbl 0827.22005) Full Text: DOI