Title: Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups Show Chinese title

Title: p进Lie群上的不变测度：p进四元数代数和p进旋转群上的哈尔积分

Abstract: We provide a general expression of the Haar measure $-$ that is, the essentially unique translation-invariant measure $-$ on a $p$-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the $p$-adic special orthogonal groups in dimension two, three and four (for every prime number $p$). In particular, the Haar measure on $\mathrm{SO}(2,\mathbb{Q}_p)$ is obtained by a direct application of our general formula. As for $\mathrm{SO}(3,\mathbb{Q}_p)$ and $\mathrm{SO}(4,\mathbb{Q}_p)$, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain $p$-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field $\mathbb{Q}_p$ and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the $p$-adic special orthogonal groups, with potential applications in $p$-adic quantum mechanics and in the recently proposed $p$-adic quantum information theory. Show Chinese abstract

Abstract: 我们给出了 Haar 测度$-$的一个一般表达式，即在一个$p$-进 Lie 群上唯一的本质上平移不变的测度$-$。 然后我们论证了这个测度可以被视为由群上的不变体积形式自然诱导出的测度，就像实数域上的标准 Lie 群情况一样。 作为一个重要的应用，接下来我们考虑确定在二维、三维和四维（对于每个素数$p$）的$p$-进特殊正交群上的 Haar 测度的问题。 特别是，通过直接应用我们的通用公式，得到了$\mathrm{SO}(2,\mathbb{Q}_p)$上的 Haar 测度。 至于$\mathrm{SO}(3,\mathbb{Q}_p)$和$\mathrm{SO}(4,\mathbb{Q}_p)$，我们证明了这两个群上的哈尔积分可以方便地提升到某些$p$-进李群上的哈尔积分，特殊正交群可以从这些李群中作为商群得到。 这一构造涉及在场$\mathbb{Q}_p$上的一个适当的四元数代数，并与实旋转群的四元数实现有相似之处。 我们的结果应该为在$p$-进特殊正交群上发展调和分析铺平道路，可能在$p$-进量子力学以及最近提出的$p$-进量子信息论中有潜在应用。