Geometric explanation for the “−1” offset in the pluriclosed root-parameter relation

Determine whether there exists a geometric explanation for the appearance of the “−1” offset in the relation x_{α+β} = x_α + x_β − 1 that characterizes left- and Ad(T)-invariant pluriclosed Hermitian metrics on a compact semisimple Lie group G with respect to a fixed maximal torus T and root system Δ^+. Specifically, explain why these pluriclosed metrics satisfy x_{α+β} = x_α + x_β − 1 for all positive roots α, β, α+β ∈ Δ^+, whereas the Hermitian structure (J_{𝔮}, g_{𝔮}) on the flag manifold G/T is Kähler precisely when x_{α+β} = x_α + x_β without the “−1” term.

Background

The paper classifies all left- and Ad(T)-invariant pluriclosed Hermitian metrics on compact semisimple Lie groups in terms of the root system associated with a chosen maximal torus T. In the irreducible case, the classification yields that, up to scaling on each simple factor, the parameters (x_α) for positive roots satisfy x_{α+β} = x_α + x_β − 1 whenever α, β, α+β ∈ Δ^+.

The authors note an intriguing contrast with the Kähler condition for the induced Hermitian structure (J_{𝔮}, g_{𝔮}) on the flag manifold G/T. On G/T, the Kähler condition is x_{α+β} = x_α + x_β (i.e., without the “−1” offset), and its solutions are precisely linear in the simple root parameters. The authors explicitly state that they do not know whether there is a hidden geometric reason underlying this discrepancy, motivating the open question.