2023/12/09 by Mark McKee, McKee, M., Angela Pasquale +3
Chemistry · Mathematics · #22E30 #Advanced Algebra and Geometry #Advanced NMR Techniques and Applications #FOS: Mathematics #Molecular spectroscopy and chirality #Primary: 22E45 #Representation Theory (math.RT) #secondary: 22E46
paper · pdf · doi:10.48550/arxiv.2312.05546
openalex publication_date 2023/12/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the dual pair (G,G')=(Ul,Ul') in the symplectic group Sp2ll'(ℝ). Fix a Weil representation of the metaplectic group Sp2ll'(ℝ). Let G and G' be the preimages of G and G' under the metaplectic cover Sp2ll'(ℝ)→ Sp2ll'(ℝ), and let Π⊗Π' be a genuine irreducible representation of G ×G'. We study the Weyl symbol fΠ⊗Π' of the (unique up to a possibly zero constant) symmetry breaking operator (SBO) intertwining the Weil representation with Π⊗Π'. This SBO coincides with the orthogonal projection of the space of the Weil representation onto its Π-isotypic component and also with the orthogonal projection onto its Π'-isotypic component. Hence fΠ⊗Π' can be computed in two different ways, one using Π and the other using Π'. By matching the results, we recover Weyl's theorem stating that Π⊗Π' occurs in the Weil representation with multiplicity at most one and we also recover the complete list of the representations Π⊗Π' occurring in Howe's correspondence.