We construct explicit local systems on the affine line in characteristic $p>2$, whose geometric monodromy groups are the finite symplectic groups ${\mathrm{Sp}}_{2n}(q)$ for all $n⩾2$, and others whose geometric monodromy groups are the special unitary groups ${\mathrm{SU}}_n(q)$ 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 ${\mathbb{G}}_m$, whose geometric monodromy groups are the finite symplectic groups ${\mathrm{Sp}}_{2n}(q)$ for any $n⩾2$, and others whose geometric monodromy groups are the finite general unitary groups ${\mathrm{GU}}_n(q)$ for any odd $n⩾3$.

中文翻译：

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有限辛群和酉群的指数和和总 Weil 表示

我们在特征的仿射线上构建显式局部系统 $磷>2$，其几何单峰群是有限辛群 ${\mathrm{斯普}}_{2n}(q)$ 对所有人 $n⩾2$, 以及其他几何单峰群是特殊酉群 ${\mathrm{苏}}_n(q)$ 对于所有奇怪的 $n⩾3$， 和 $q$ 任何权力 $磷$，在他们的总 Weil 表示中。这些局部系统的一个主要优点是它们相关的迹函数是一种非常简单的指数和的单参数族，即易于记忆的形式。我们还展示了超几何滑轮${\mathbb{G}}_{米}$，其几何单峰群是有限辛群 ${\mathrm{斯普}}_{2n}(q)$ 对于任何 $n⩾2$，以及其他几何单调群是有限一般酉群 ${\mathrm{顾}}_n(q)$ 对于任何奇怪的 $n⩾3$.