The characteristic masses of Niemeier lattices

Chenevier, Gaëtan

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 545-583.

Soient $L$ un réseau entier d’un espace euclidien $E$ de dimension $n$ et $W$ une représentation irréductible du groupe orthogonal de $E$. Nous donnons un algorithme calculant la dimension du sous-espace des éléments de $W$ invariants par le groupe $\mathrm{O}\left(L\right)$ des isométries de $L$. Une étape clef est de déterminer, pour tout polynôme $P$, la proportion des éléments de $\mathrm{O}\left(L\right)$ de polynôme caractéristique $P$, une collection de nombres rationnels que nous appelons les masses caractéristiques de $L$. En guise d’application, nous déterminons les masses caractéristiques de tous les réseaux de Niemeier, et plus généralement de tous les réseaux pairs de déterminant $\le 2$ en dimension $n\le 25$.

Pour les réseaux de Niemeier, en guise de vérification, nous donnons une méthode alternative (et humaine) pour calculer leurs masses caractéristiques. L’ingrédient principal est la détermination, pour chaque réseau de Niemeier $L$ de système de racines $R$ non vide, des $\mathrm{G}\left(R\right)$-classes de conjugaison d’éléments du sous-groupe « ombral » $\mathrm{O}\left(L\right)/\mathrm{W}\left(R\right)$ de $\mathrm{G}\left(R\right)$, où $\mathrm{G}\left(R\right)$ est le groupe des automorphismes du diagramme de Dynkin de $R$, et $\mathrm{W}\left(R\right)$ son groupe de Weyl.

Ces résultats ont des applications à l’étude des espaces de formes automorphes des groupes orthogonaux de formes quadratiques sur $\mathbb{Q}$ définies positives : nous donnons des formules concrètes pour la dimension de ces espaces en niveau $1$, comme fonction du poids $W$, en tout rang $n\le 25$.

Let $L$ be an integral lattice in an $n$-dimensional Euclidean space $E$ and $W$ an irreducible representation of the orthogonal group of $E$. We give an implemented algorithm computing the dimension of the subspace of invariants in $W$ under the isometry group $\mathrm{O}\left(L\right)$ of $L$. A key step is the determination, for any polynomial $P$, of the proportion of elements in $\mathrm{O}\left(L\right)$ with characteristic polynomial $P$, a collection of rational numbers that we call the 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 $\le 2$ in dimension $n\le 25$.

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 $\mathrm{G}\left(R\right)$-conjugacy classes of the elements of the “umbral” subgroup $\mathrm{O}\left(L\right)/\mathrm{W}\left(R\right)$ of $\mathrm{G}\left(R\right)$, where $\mathrm{G}\left(R\right)$ is the automorphism group of the Dynkin diagram of $R$, and $\mathrm{W}\left(R\right)$ 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 $n=25$.

Classification : 11F, 11F55, 11H55, 11H56, 11H71, 20D08, 22C05
Mots clés : Euclidean lattices, isometry groups, Niemeier lattices, automorphic forms

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