Solvable Groups, Free Divisors and Nonisolated Matrix Singularities I: Towers of Free Divisors

Damon, James; Pike, Brian

Annales de l'Institut Fourier, Tome 65 (2015) no. 3, pp. 1251-1300.

Nous introduisons une méthode pour obtenir des nouvelles classes de diviseurs libres à partir de représentations $V$ de groupes algébriques linéaires connexes $G$ pour lesquelles $dimG=dimV$ et $V$ a une orbite ouverte. Nous donnons des conditions suffisantes pour lesquelles le complémentaire de cette orbite ouverte, la « variété des orbites exceptionelles », est une diviseur libre (ou un diviseur libre* plus faible) pour des « représentations par blocs » à la fois des groupes solvables et des extensions des groupes réductifs par ces groupes. Ce sont des représentations pour lesquelles la matrice définie à partir d’une base des « champs des vecteurs associés » de la représentation $V$, a une forme triangulaire bloc et les blocs satisfont certaines conditions de non-singularité.

Pour les tours de groupes de Lie et leurs représentations ce résultat donne une tour de diviseurs libres obtenue en avoisinant successivement des variétés de matrices singulières. Il s’applique aux groupes solvables qui donnent la factorisation classique du type Cholesky et une forme modifiée de celle ci, sur les espaces des matrices $m\times m$ symétriques, antisymétriques, ou générales. Pour les matrices antisymétriques, il s’étend aussi aux représentations des algèbres de Lie solvables et non-linéaires de dimension infinie.

We introduce a method for obtaining new classes of free divisors from representations $V$ of connected linear algebraic groups $G$ where $dimG=dimV$, with $V$ having an open orbit. We give sufficient conditions that the complement of this open orbit, the “exceptional orbit variety”, is a free divisor (or a slightly weaker free* divisor) for “block representations” of both solvable groups and extensions of reductive groups by them. These are representations for which the matrix defined from a basis of associated “representation vector fields” on $V$ has block triangular form, with blocks satisfying certain nonsingularity conditions.

For towers of Lie groups and representations this yields a tower of free divisors, successively obtained by adjoining varieties of singular matrices. This applies to solvable groups which give classical Cholesky-type factorization, and a modified form of it, on spaces of $m\times m$ symmetric, skew-symmetric or general matrices. For skew-symmetric matrices, it further extends to representations of nonlinear infinite dimensional solvable Lie algebras.

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