On the notion of the parabolic and the cuspidal support of smooth-automorphic forms and smooth-automorphic representations

In this paper we describe several new aspects of the foundations of the representation theory of the space of smooth-automorphic forms (i.e., not necessarily $K_\infty$-finite automorphic forms) for general connected reductive groups over number fields. Our role model for this space of smooth-automorphic forms is a ''smooth version'' of the space of automorphic forms, whose internal structure was the topic of a famous paper of Franke. We prove that the important decomposition along the parabolic support, and the even finer - and structurally more important - decomposition along the cuspidal support of automorphic forms transfer in a topologized version to the larger setting of smooth-automorphic forms. In this way, we establish smooth-automorphic versions of the main results of a paper of Franke-Schwermer and of Moeglin-Waldspurger's book, III.2.6.