Hypergeometric sheaves and finite symplectic and unitary groups

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 A¹, whose geometric monodromy groups are Sp_2n(q), respectively SU_n(q), in their total Weil representation of degree qⁿ, 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.