DUALITY IN THE GROTHENDIECK GROUP OF THE CATEGORY OF FINITE-LENGTH SMOOTH REPRESENTATIONS OF A P-ADIC REDUCTIVE GROUP

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.