The Aubert and Bernstein involutions for disconnected groups

We extend to arbitrary disconnected reductive $p$-adic groups the duality on the category of smooth finite-length complex representations defined by Aubert, as well as its cohomological analog defined by Bernstein, and prove various properties of these functors, such as uniqueness, preservation of irreducibility, compatibility with parabolic induction and restriction, and a character formula. As a sample application, we obtain a definition of the Steinberg representation for a disconnected reductive $p$-adic group, compute its character, and discuss its twisted endoscopic properties.