Automorphic $C^*$-algebras of reductive groups

Abstract

Let $G$ be a reductive group over a number field $F$. We define $C^*_{\rm aut}(G(\mathbb A))$, the automorphic $C^*$-algebra of $G$, to be the $C^*$-algebraic image of the full automorphic representation of $G(\mathbb A)$ on $L^2(G(F)\backslash G(\mathbb A))$. Using the Langlands spectral decomposition with respect to discrete Levi data, we construct an injective $*$-homomorphism \[ C^*_{\rm aut}(G(\mathbb A)) \longrightarrow \bigoplus_{[M,σ]} K_{C_0(\widehat{A_M})} \left( \operatorname{Ind}_P^G C_0(\widehat{A_M},H_{M,σ}) \right)^{W(G,M,σ)}, \] where $[M,σ]$ ranges over the associate classes of discrete Levi data. When $G$ is $\operatorname{GL}(n)$ or an inner form of it, we prove that this map is an isomorphism of $C^*$-algebras.