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

2021/08/13 by Grobner, Harald, Žunar, Sonja · 1 citation
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2108.06369

Abstract

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_∞-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.

Cited by

• Factorization of periods, construction of automorphic motives and Deligne's conjecture over CM-fields

Related

• Cuspidal representations of reductive groups
• Smooth cuspidal automorphic forms and integrable discrete series
• Congruences of algebraic automorphic forms and supercuspidal\n representations
• Galois deformation spaces with a sparsity of automorphic points
• Representations of field automorphism groups