标题： 一个关于类似Leech格的引理 显示英文标题

标题： A Lemma on Leech-like Lattices

摘要： 一个Leech对被定义为一对$(G,S)$，其中$S$是一个没有根的正定偶格子，配备有限群$G$的忠实作用，使得在$S$作用下$G$的不变子格是平凡的，并且$G$在$S$的判别群上的诱导作用也是平凡的。 这种结构在研究超凯勒流形和作用在它们上的辛自同构时自然出现。 一个由Gaberdiel--Hohenegger--Volpato提出的重要的引理断言，若$(G,S)$是一个Leech对，则当$rank(S)+\ell(A_S)\le 24$时，它可以被原始嵌入到Leech格中。 然而，Marquand和Muller给出的反例表明，原始证明是不完整的。 在本文中，我们修改了原始方法，以提供该引理的完整且具有概念性的证明。 显示英文摘要

摘要： A Leech pair is defined as a pair $(G,S)$, where $S$ is a positive definite even lattice without roots, equipped with a faithful action of a finite group $G$, such that the invariant sublattice of $S$ under the action of $G$ is trivial, and the induced action of $G$ on the discriminant group of $S$ is also trivial. This structure appears naturally when investigating hyperk\"ahler manifolds and the symplectic automorphisms acting on them. An important lemma due to Gaberdiel--Hohenegger--Volpato asserts that a Leech pair $(G,S)$ admits a primitive embedding into the Leech lattice if $rank(S)+\ell(A_S)\le 24$. However, the original proof is incomplete, as demonstrated by a counterexample given by Marquand and Muller. In this paper, we modify the original approach to provide a complete and conceptual proof of the lemma.