We define a motivic analogue of the Haar measure for groups of the form$G(kllp t rp)$, where~$k$ is an algebraically closed fieldof characteristic zero, and $G$ is a reductive algebraic group defined over$k$.A classical Haar measure on such groups does notexist since they are not locally compact.We use the theory of motivic integration introduced by M.~Kontsevich todefine an additive function on a certain natural Boolean algebra of subsets of$G(kllp t rp)$. This function takes values in the so-called dimensionalcompletion ofthe Grothendieck ring of the category of varieties over the basefield. It is invariant under translations by all elements of $G(kllp t rp)$,and therefore we call it a motivic analogue of Haar measure.We give an explicit construction of the motivic Haar measure, and then provethat the result is independent of all the choices that are made in the process.
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机译：我们为形式为$ G（kllp t rp）$的组定义Haar测度的动机类似物，其中〜$ k $是特征为零的代数封闭字段，而$ G $是定义为超过$ k $的归约代数组由于此类群不是局部紧凑的，因此不存在经典的Haar测度。我们使用M.〜Kontsevich引入的动机积分理论，在$ G（kllp t rp）子集的某些自然布尔代数上定义加法函数$。该函数采用基场上品种类别的所谓格罗腾迪克环的维完成度中的值。 $ G（kllp t rp）$的所有元素在翻译下都是不变的，因此我们称其为Haar测度的动机类似物。我们给出了动机Haar测度的显式构造，然后证明了结果独立于所有在过程中做出的选择。
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