Exponential sums and total Weil representations of finite symplectic and unitary groups

We construct explicit local systems on the affine line in characteristic (Formula presented.), whose geometric monodromy groups are the finite symplectic groups (Formula presented.) for all (Formula presented.), and others whose geometric monodromy groups are the special unitary groups (Formula presented.) for all odd (Formula presented.), and (Formula presented.) any power of (Formula presented.), in their total Weil representations. One principal merit of these local systems is that their associated trace functions are one-parameter families of exponential sums of a very simple, that is, easy to remember, form. We also exhibit hypergeometric sheaves on (Formula presented.), whose geometric monodromy groups are the finite symplectic groups (Formula presented.) for any (Formula presented.), and others whose geometric monodromy groups are the finite general unitary groups (Formula presented.) for any odd (Formula presented.).