Harmonic Analysis on the quotient $\Q^\times\ltimes\Q\bs\A^1\ltimes \A$

Abstract

Let $G$ be the semidirect product $\A^1\ltimes \A$ of the adeles and the norm 1 ideles. Let $\Ga$ be the discrete subgroup $\Q^\times\ltimes\Q$. In this paper the trace formula for this setting is established and used to give the complete decomposition of the $G$-representation on $L^2(\Ga\bs G)$. It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.