## Abstract

We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups Sp_{2n}(q) for any odd n ≥ 3, for q any power of an odd prime p. We construct other hypergeometric sheaves whose geometric monodromy groups are the finite unitary groups GU_{n}(q), for any even n ≥ 2, for q any power of any prime p. Suitable Kummer pullbacks of these sheaves yield local systems on A^{1}, whose geometric monodromy groups are Sp_{2n}(q), respectively 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 GU_{n}(q) withn 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.

Original language | English (US) |
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Pages (from-to) | 577-691 |

Number of pages | 115 |

Journal | Cambridge Journal of Mathematics |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - 2021 |

## All Science Journal Classification (ASJC) codes

- General Mathematics

## Keywords

- Finite simple groups
- Hypergeometric sheaves
- Local systems
- Monodromy groups
- Weil representations