Langlands Reciprocity for C ^∗-Algebras

Abstract

We introduce a C ^∗-algebra ${\mathcal A}_V$ of a variety V over the number field K and a C ^∗-algebra ${\mathcal A}_G$ of a reductive group G over the ring of adeles of K. Using Pimsner’s Theorem, we construct an embedding ${\mathcal A}_V\hookrightarrow {\mathcal A}_G$, where V is a G-coherent variety, e.g. the Shimura variety of G. The embedding is an analog of the Langlands reciprocity for C ^∗-algebras. It follows from the K-theory of the inclusion ${\mathcal A}_V\subset {\mathcal A}_G$ that the Hasse-Weil L-function of V is a product of the automorphic L-functions corresponding to irreducible representations of the group G.