Generalizing Lusztig's total positivity

Olivier Guichard (1) , Anna Wienhard (2)

1 IRMA - Institut de Recherche Mathématique Avancée
2 MPIM - Max Planck Institute for Mathematics

We introduce the notion of Θ-positivity in real simple Lie groups. This notion at the same time generalizes Lusztig's total positivity in split real Lie groups and invariant orders in Lie groups of Hermitian type. We show that there are four families of Lie groups which admit Θ-positive structures, and investigate basic properties of Θ-positivity.

Domaines: Géométrie différentielle [math.DG]; Théorie des groupes [math.GR]

Olivier Guichard, Anna Wienhard. Generalizing Lusztig's total positivity. 2024. ⟨hal-03753918v2⟩