Hypergeometric sheaves and finite symplectic and unitary groups
type: Item
instance of: scholarly article
title: Hypergeometric sheaves and finite symplectic and unitary groups
zbMATH Open document ID: 1486.11150
author: Nicholas M. Katz
author: Pham Huu Tiep
published in: Cambridge Journal of Mathematics
publication date: 2022-01-07T00:00:00Z
review text: The article belongs to a series of papers by the authors in which they exhibit easy-to-remember exponential sums in similar contexts. In this paper they construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups \(\mathrm{Sp}_{2n}(q)\) for any odd \(n\geq 3\), where \(q\) is any power of the odd prime \(p\). They construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups \(\mathrm{GU}_n(q)\), for any even \(n\geq 2\). Suitable Kummer pullbacks of these sheaves yield local systems on \(\mathbb{A}^1\), whose geometric monodromy groups are \(\mathrm{Sp}_{2n}(q)\) and \(\mathrm{SU}_n(q)\) respectively, in their total Weil representation of degree \(q^n\), and whose trace functions are one-parameter families of two-variable exponential sums. The authors treat for the first time unitary groups \(\mathrm{GU}_n(q)\) with \(n\) even via hypergeometric sheaves. They use a maximal torus which is a product of two sub-tori to furnish a generator of local monodromy at 0 both in the symplectic and the unitary cases. Families of two-variable exponential sums occur for the first time in the context of finite classical groups.
Mathematics Subject Classification ID: 11T23
Mathematics Subject Classification ID: 14D05
Mathematics Subject Classification ID: 20C33
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 7453334
zbMATH Keywords: local systems
zbMATH Keywords: hypergeometric sheaves
zbMATH Keywords: monodromy groups
zbMATH Keywords: finite simple groups
zbMATH Keywords: Weil representation
reviewed by: Vladimir P. Kostov
MaRDI profile type: Publication
DOI: 10.4310/CJM.2021.V9.N3.A2
description: scientific article; zbMATH DE number 7453334