Isotropic Hermite invariant and orthogonal Lorentzian lattices (Q863618)

\textit{A. M. Bergé} and \textit{J. Martinet} [Enseign. Math., II. Sér. 41, No. 3--4, 335--365 (1995; Zbl 0848.52006)] introduced the notion of an orthogonal lattice, i.e. a lattice \(\Lambda\) in a Euclidean space \(E\) together with an isometry \(\sigma\neq \pm \text{Id}\) of order \(2\) mapping \(\Lambda\) onto its dual. Orthogonal lattices \((\Lambda,\sigma)\) carry an underlying structure of indefinite integral bilinear \({\mathbb Z}\)-module with respect to the bilinear form \(\alpha_\Lambda(u,v)=\langle u,\sigma(v)\rangle\) and are characterized by its type (even or odd) and its signature \((p,q)\). The well-known classical Hermite invariant of \(\Lambda\) is defined by \(\mu (\Lambda)=\min\{ | u| ^2\mid u\in\Lambda\setminus\{0\}\}\). In the present paper, the author introduces refined versions \(\mu^k(\Lambda)\) of the Hermite invariant for orthogonal lattices and for \(k\in {\mathbb Z}\) (resp. \(k\in 2{\mathbb Z}\)) for odd (resp. even) \((\Lambda,\alpha_\Lambda)\), defined as follows: \[ \mu^k(\Lambda)=\min\{ | u| ^2\mid u\in \Lambda\setminus\{0\},\;\alpha (u,u)=k\}. \] \(\mu^0\) is called the isotropic hermite invariant. The main focus is on orthogonal lattices of signature \((n,1)\) which the author calls Lorentzian orthogonal lattices. In section 1, he computes their maximal density up to dimension \(12\) and also in dimension \(18\) in the even case. The methods are of a geometric nature and are based on identifying orthogonal lattices with their underlying bilinear structure \(\alpha_\Lambda\) (and up to isometry) with the quotient of a symmetric space \(X\) by a discrete group \(\Theta\) acting on \(X\). In the case of Lorentzian lattices, \(X\) can be naturally identified with the hyperbolic space \textbf{H}\(^n\). This geometric approach for orthogonal lattices of arbitrary signature is developed in section 2. In the final section 3, the author returns to Lorentzian orthogonal lattices and develops a Voronoï theory for the isotropic Hermite invariant \(\mu^0\), including a Voronoï-type algorithm for classifying the local maxima of \(\mu^0\).