  We construct explicit local systems on the affine line in characteristic
$p>2$, whose geometric monodromy groups are the finite symplectic groups
$Sp_{2n}(q)$ for all $n \ge 2$, and others whose geometric monodromy groups are
the special unitary groups $SU_n(q)$ for all odd $n \ge 3$, and $q$ any power
of $p$, 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, i.e., easy to remember, form. We also
exhibit hypergeometric sheaves on $G_m$, whose geometric monodromy groups are
the finite symplectic groups $Sp_{2n}(q)$ for any $n \ge 2$, and others whose
geometric monodromy groups are the finite general unitary groups $GU_n(q)$ for
any odd $n \geq 3$.
