2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/154833After explaining the concepts of Langlands dual and miniscule representations, we define an analog of the Gauss sum for any compact, simple Lie group with a simply laced Lie algebra. We then show a reciprocity property when a Lie group is exchanged with its Langlands dual. We also explore the relation with theta functions and modular transformations. In the non-simply laced case, we construct a unitary representation of the Hecke group which involves interesting new phase factors.12 pages, corrected versionRepresentation Theory22E46; 11F03; 11L03text