2024/10/08 by Marcelo De Martino, De Martino, Marcelo, Eric Opdam +1
Mathematics · Physics and Astronomy · #11R34 #20J06 #22E55 #Advanced Algebra and Geometry #Advanced Differential Geometry Research #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT) #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2410.06346
openalex publication_date 2024/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For an algebraic torus defined over a local (or global) field F, a celebrated result of R.P. Langlands establishes a natural homomorphism from the group of continuous cohomology classes of the Weil group, valued in the dual torus, onto the space of complex characters of the rational points of the torus (or automorphic characters in the global case). We expand on this result by detailing its topological aspects. We show that if we topologize the relevant spaces of continuous homomorphisms and continuous cochains using the compact-open topology, Langlands's map becomes a (surjective, finite-to-one) homomorphism of abelian complex Lie groups. Moreover, we demonstrate that, in both the local and global settings, the subset of unramified characters is the identity component of the relevant space of characters. Finally, we compare the group of unramified characters with the Galois (co)invariants of the dual torus.