The Characteristic Masses of Niemeier Lattices

Abstract

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 $n=25$.