2023/12/19 by Gilbert Moss, Moss, Gilbert, Justin Trias +1 · 1 citation
Mathematics · #11F27 #11S23 #20C20 #22E50 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2312.12031
openalex publication_date 2023/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let R be a commutative ℤ[1/p]-algebra, let m ≤ n be positive integers, and let Gn=GLn(F) and Gm=GLm(F) where F is a p-adic field. The Weil representation is the smooth R[Gn× Gm]-module Cc∞(Matn× m(F),R) with the action induced by matrix multiplication. When R=ℂ or is any algebraically closed field of banal characteristic compared to Gn and Gm, the local theta correspondence holds by the work of Howe and Mínguez. At the level of supercuspidal support, we interpret the theta correspondence as a morphism of varieties θR, which we describe as an explicit closed immersion. For arbitrary R, we construct a canonical ring homomorphism θ^#R : \mathfrakZR(Gn)→ \mathfrakZR(Gm) that controls the action of the center \mathfrakZR(Gn) of the category of smooth R[Gn]-modules on the Weil representation. We use the rank filtration of the Weil representation to first obtain θℤ[1/p]^#, then obtain θ^#R for arbitrary R by proving \mathfrakZR(Gn) is compatible with scalar extension. In particular, the map Spec(\mathfrakZR(Gm))→ Spec(\mathfrakZR(Gn)) induced by θR^# recovers θR in the R=ℂ case and in the banal case. We use gamma factors to prove θR^# is surjective for any R. Finally, we describe θ^#R in terms of the moduli space of Langlands parameters and use this description to give an alternative proof of surjectivity in the tamely ramified case.