2023/12/04 by Itay Glazer, Julia Gordon, Glazer, Itay +3 · 1 citation
Mathematics · #14B05 #17B08 #20G05 #20G05 (Primary) 14N20 #22E30 #22E35 #22E46 #32S22 #43A30 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2312.01591
openalex publication_date 2023/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a reductive group over a local field F of characteristic 0. By Harish-Chandra's regularity theorem, the character Θπ of an irreducible, admissible representation π of G is given by a locally integrable function θπ on G. It is a natural question whether θπ has better integrability properties, namely, whether it is locally L1+ε-integrable for some ε>0. It turns out that the answer is positive, and this gives rise to a new singularity invariant of representations ε⋆(π):=sup\ ε:θπ∈ LLoc1+ε(G)\ , which we explore in this paper. We provide a lower bound on ε⋆(π) which depends only on the absolute root system of G, and explicitly determine ε⋆(π) in the case of a p-adic GLn. This is done by studying integrability properties of the Fourier transforms \widehatξO of stable Richardson nilpotent orbital integrals ξO. We express ε⋆(\widehatξO) as the log-canonical threshold of a suitable relative Weyl discriminant, and use a resolution of singularities algorithm coming from the theory of hyperplane arrangements, to compute it in terms of the partition associated with the orbit. We obtain several applications; firstly, we provide bounds on the multiplicities of K-types in irreducible representations of G in the p-adic case, where K is an open compact subgroup. We further obtain bounds on the multiplicities of the irreducible representations appearing in the space L2(K/L), where K is a compact simple Lie group, and L≤ K is a Levi subgroup. Finally, we discover surprising applications in random matrix theory, namely to the study of the eigenvalue distribution of powers of random unitary matrices.