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

Aubert, Anne-Marie

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.

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