Contents Online

# Cambridge Journal of Mathematics

## Volume 9 (2021)

### Number 3

### Hypergeometric sheaves and finite symplectic and unitary groups

Pages: 577 – 691

DOI: https://dx.doi.org/10.4310/CJM.2021.v9.n3.a2

#### Authors

#### Abstract

We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups $\mathrm{Sp}_{2n} (q)$ for any **odd** $n \geq 3$, for $q$ any power of an odd prime $p$. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups $\mathrm{GU}_n (q)$, for any **even** $n \geq 2$, for $q$ any power of any prime $p$. Suitable Kummer pullbacks of these sheaves yield local systems on $\mathbb{A}^1$, whose geometric monodromy groups are $\mathrm{Sp}_{2n} (q)$, respectively $\mathrm{SU}_n (q)$, in their total Weil representation of degree $q^n$, and whose trace functions are simple-to-remember one-parameter families of two-variable exponential sums. The main novelty of this paper is three-fold. First, it treats unitary groups $\mathrm{GU}_n (q)$ with $n$ **even** via hypergeometric sheaves for the first time. Second, in both the symplectic and the unitary cases, it uses a maximal torus which is a product of two sub-tori to furnish a generator of local monodromy at $0$. Third, this is the first natural occurrence of families of two-variable exponential sums in the context of finite classical groups.

#### Keywords

local systems, hypergeometric sheaves, monodromy groups, finite simple groups, Weil representation

#### 2010 Mathematics Subject Classification

Primary 11T23, 14D05, 20C33. Secondary 20C15, 20D06, 20G40.

The second author gratefully acknowledges the support of the NSF (grant DMS-1840702), the Joshua Barlaz Chair in Mathematics, and the Charles Simonyi Endowment at the Institute for Advanced Study (Princeton).

Received 30 July 2021

Published 7 December 2021