AI 中文总结
研究非阿贝尔幂零格的非交换弗洛凯 - 布洛赫理论，通过构建表示论部分的布洛赫型替代，证明\(\pi_l|_\Gamma\)的精确限制定理，在有限维有理纤维上构造普兰切尔测度，给出幂零格相关恒等式并恢复皮特利克公式。
AI 中文摘要
经典的弗洛凯 - 布洛赫理论在格的特征环面上分解阿贝尔周期问题。对于非阿贝尔幂零格，非I型障碍排除了全酉对偶的可比参数化。我们不试图消除此障碍，而是在理论的表示论部分构建一个精确的布洛赫型替代，它可从有理基里洛夫数据和有限维有理纤维中看到。设\(\Gamma\)是无挠有限生成幂零群，\(G\)是其马尔采夫完备化。对于与有理基里洛夫参数\(l\in\mathfrak g_{\mathbb Q}^{*}\)相关的\(G\)的不可约酉表示\(\pi_l\)，我们证明了\(\pi_l|_\Gamma\)的精确限制定理。分支首先由与有理极化相关的诱导表示描述。在与有限维表示相关的有理奇数轨迹上，它进一步分解为\(\Gamma\)的有限维不可约表示。在这些有限维有理纤维上，我们构造了一个正的有限可加普兰切尔测度。它给出了幂零格的傅里叶反演和归一化迹恒等式，在离散海森堡情形下恢复了皮特利克公式。
英文摘要
Classical Floquet-Bloch theory decomposes abelian periodic problems over the character torus of the lattice. For nonabelian nilpotent lattices, the non-type I obstruction rules out a comparable parametrization of the full unitary dual. We do not attempt to remove this obstruction. Instead, we construct an exact Bloch-type replacement on the representation-theoretic part of the theory which is visible from rational Kirillov data and from finite-dimensional rational fibers. Let $Γ$ be a torsion-free finitely generated nilpotent group and let $G$ be its Malcev completion. For an irreducible unitary representation $π_l$ of $G$ attached to a rational Kirillov parameter $l\in\mathfrak{g}_{\mathbb Q}^{*}$, we prove an exact restriction theorem for $π_l|_Γ$. The branching is first described by induced representations attached to rational polarizations. On the rational odd locus relevant to finite-dimensional representations, it further decomposes into finite-dimensional irreducible representations of $Γ$. On these finite-dimensional rational fibers we construct a positive finitely additive Plancherel measure. It gives Fourier inversion and normalized trace identities for nilpotent lattices, recovering Pytlik's formula in the discrete Heisenberg case.
Comments60 pages. v2: restored acknowledgements inadvertently omitted; corrected metadata for the title and abstract; made minor clarifications in the exposition. Main results unchanged. Self-contained representation-theoretic foundations paper; analytic applications are in the companion paper arXiv:2607.13890. Draws on and substantially revises the corresponding part of arXiv:2509.16848
