Hypergeometric sheaves and finite symplectic and unitary groups

Abstract

We construct hypergeometric sheaves whose geometric monodromy groups are the finite symplectic groups Sp2n(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 GUn(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, whose geometric monodromy groups are Sp2n(q), respectively SUn(q), in their total Weil representation of degree q, and whose trace functions are simple-to-remember one-parameter families of two-variable exponential sums. The main novelty of this paper is two-fold. First, it treats unitary groups GUn(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.