Hypergeometric sheaves and finite symplectic and unitary groups (Q2065054)

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.