论文标题
Niemeier晶格的特征质量
The Characteristic Masses of Niemeier Lattices
论文作者
论文摘要
令$ l $为Euclidean Space $ \ Mathbb {r}^n $和$ w $的整体晶格，$ \ Mathbb {r}^n $的正交组的不可避免表示。我们给出了一个实施的算法计算$ w $ $ w $下不变的子空间的尺寸，一个关键步骤是确定$ {\ rm o}（l）中具有任何给定特征多项式的元素数量，我们称为$ l $的{\ it特征质量}的基准。作为一个应用程序，我们确定了所有Niemeier晶格的特征性质量，以及更普遍的均匀的$ \ leq 2 $ in dimension $ n \ leq 25 $的均匀晶格。 对于Niemeier晶格，作为验证，我们提供了特征性质量的替代（人）计算。主要成分是确定，对于每个Niemeier晶格$ l $，带有非空根系统$ r $，是$ {\ rm g}（r）$ - $ - $ - conjugacy类的“ umbral” subgroup $ {\ rm o}（\ rm o}（lm o}（l）/{\ rm w} $ g rm g g rm g rm g rm g rm g rm g rm g rm of conjugacy类。 $ {\ rm g}（r）$是$ r $的dynkin图的自动形态组，而$ {\ rm w}（r）$ weyl group。 这些结果对研究$ \ Mathbb {q} $的$ n $变量的确定正交组的自动形式的空间有后果。例如，我们在$ 1 $ case中提供具体的尺寸公式，作为权重$ W $的函数，最多为dimension $ n = 25 $。
Let $L$ be an integral lattice in the Euclidean space $\mathbb{R}^n$ and $W$ an irreducible representation of the orthogonal group of $\mathbb{R}^n$. We give an implemented algorithm computing the dimension of the subspace of invariants in $W$ under the isometry group ${\rm O}(L)$ of $L$. A key step is the determination of the number of elements in ${\rm O}(L)$ having any given characteristic polynomial, a datum that we call the {\it characteristic masses} of $L$. As an application, we determine the characteristic masses of all the Niemeier lattices, and more generally of any even lattice of determinant $\leq 2$ in dimension $n \leq 25$. For Niemeier lattices, as a verification, we provide an alternative (human) computation of the characteristic masses. The main ingredient is the determination, for each Niemeier lattice $L$ with non-empty root system $R$, of the ${\rm G}(R)$-conjugacy classes of the elements of the "umbral" subgroup ${\rm O}(L)/{\rm W}(R)$ of ${\rm G}(R)$, where ${\rm G}(R)$ is the automorphism group of the Dynkin diagram of $R$, and ${\rm W}(R)$ its Weyl group. These results have consequences for the study of the spaces of automorphic forms of the definite orthogonal groups in $n$ variables over $\mathbb{Q}$. As an example, we provide concrete dimension formulas in the level $1$ case, as a function of the weight $W$, up to dimension $n=25$.
