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We construct explicit local systems on the affine line in characteristic    p  >  2   , whose geometric monodromy groups are the finite symplectic groupsSp   2  n(  q  )for all    n  ⩾  2   , and others whose geometric monodromy groups are the special unitary groupsSU  n    (  q  )for all odd    n  ⩾  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, that is, easy to remember, form. We also exhibit hypergeometric sheaves on    G  m   , whose geometric monodromy groups are the finite symplectic groupsSp   2  n(  q  )for any    n  ⩾  2   , and others whose geometric monodromy groups are the finite general unitary groupsGU  n    (  q  )for any odd    n  ⩾  3   .

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