2015/04/14 by Yongqi Feng, Feng, Yongqi, Eric Opdam +1
Chemistry · Mathematics · #20C08 (Primary) #22D25 #43A30 (Secondary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Molecular spectroscopy and chirality #Representation Theory (math.RT) #math.RT #msc:20C08 #msc:22D25 #msc:43A30
paper · pdf · doi:10.48550/arxiv.1504.03458
56 pages; accepted by Advances in Mathematics
openalex publication_date 2015/04/14 · arxiv created 2020/09/07 · arxiv updated 2020/09/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The formal degree of a unipotent discrete series character of a simple linear algebraic group over a non-archimedean local field (in the sense of Lusztig), is a rational function of the cardinality q of the residue field. The irreducible factors of this rational function are q and cyclotomic polynomials. We prove that the formal degree of a supercuspidal unipotent representation determines its Lusztig-Langlands parameter, up to twisting by weakly unramified characters. For split exceptional groups this result follows from the work of Mark Reeder, and for the remaining exceptional cases this is verified by the first name author in arXiv:1708.09547. In the present paper we treat the classical families. The main result of this article characterizes unramified Lusztig-Langlands parameters which support a cuspidal local system in terms of formal degrees. The result implies the uniqueness of so-called cuspidal spectral transfer morphisms (as introduced in arXiv:1310.7193) between unipotent affine Hecke algebras (up to twisting by unramified characters). In arXiv:1310.7790 the essential uniqueness of arbitrary unipotent spectral transfer morphisms was reduced to the cuspidal case.