Generalizing Lusztig’s total positivity

Abstract We introduce the notion of $\Theta $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> -positivity in real semisimple 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 simple Lie groups which admit a positive structure relative to a subset $\Theta$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> of simple roots, and investigate fundamental properties of $\Theta $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Θ</mml:mi> </mml:math> -positivity. We define and describe the positive and nonnegative unipotent semigroups and show that they give rise to a notion of positive $n$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> </mml:math> -tuples in flag varieties.