Generalizing Lusztig's total positivity

TL;DR

This paper introduces $ heta$-positivity in real simple Lie groups, unifying Lusztig's total positivity and invariant orders, and identifies four families of Lie groups with this structure.

Contribution

It generalizes Lusztig's total positivity to a broader class of Lie groups and classifies those admitting $ heta$-positive structures.

Findings

Four families of Lie groups admit $ heta$-positivity.

$ heta$-positivity generalizes Lusztig's total positivity.

Basic properties of $ heta$-positivity are established.

Abstract

We introduce the notion of $\Theta$-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 $\Theta$-positive structures, and investigate basic properties of $\Theta$-positivity.