Let $U_1,U_2$ be connected commutative unipotent algebraic groups defined over an algebraically closed field k of characteristic $p>0$ and let $\mathcal{L}$ be a bimultiplicative ${\bar{\mathbb{Q}}}_{\ell }$-local system on $U_1\times U_2$. In this paper we will study the ${\bar{\mathbb{Q}}}_{\ell }$-cohomology $H_c^{⁎}(U_1\times U_2,\mathcal{L})$, which turns out to be supported in only one degree. We will construct a finite Heisenberg group Γ which naturally acts on $H_c^{⁎}(U_1\times U_2,\mathcal{L})$ as an irreducible representation. We will give two explicit realizations of this cohomology and describe the relationship between these two realizations as a finite Fourier transform.

中文翻译：

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单能群上双乘局部系统的同调

让 ${ü}_{\mathrm{1个}}，{ü}_2$被连接在特征为代数的封闭域k上定义的交换单能代数群$p>0$ 然后让 $\mathcal{大号}$ 成为二乘法 ${\stackrel{〜}{\mathbb{问}}}_{\ell }$-本地系统上 ${ü}_{\mathrm{1个}}\times {ü}_2$。在本文中，我们将研究${\stackrel{〜}{\mathbb{问}}}_{\ell }$-同调 $H_C^{⁎}（{ü}_{\mathrm{1个}}\times {ü}_2，\mathcal{大号}）$，结果证明只有一个程度的支持。我们将构建一个自然作用于的有限海森堡群Γ$H_C^{⁎}（{ü}_{\mathrm{1个}}\times {ü}_2，\mathcal{大号}）$作为不可简化的表示。我们将给出该同调的两个显式实现，并将这两个实现之间的关系描述为有限傅立叶变换。