# zbMATH — the first resource for mathematics

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
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
Full Text: