Linearity of the holomorphic function h′α associated to extremal sets

Determine whether there exists an index α in the family of extremal functions {hα} on the irreducible bounded symmetric domain Ω ⊂ C^N such that the holomorphic function h′α on C^N—constructed as a C-linear combination of the functions H1,…,HN′ appearing in the polynomial expansion ho(z,ζ) = 1 + Σ_{ℓ=1}^{N′} Hℓ(z)Hℓ(−ζ)—is actually linear. Here H_j(z) = z_j for 1 ≤ j ≤ N and H_ℓ(z) = G_{ℓ−N}(z) for N+1 ≤ ℓ ≤ N′, with G_i homogeneous polynomials of degree at least 2; h′α is defined by the property Zero(hα) = Zero(|h′α|^2) that arises when the system of functional equations of a holomorphic isometry F : (D, g_D) → (Ω, g_Ω) is not sufficiently non-degenerate.

Background

In the setting of holomorphic isometries between bounded symmetric domains with respect to Bergman metrics, Mok showed that if the system of functional equations of an isometry F is not sufficiently non-degenerate, then there exists a family of extremal functions {hα} on the target domain Ω whose common zero set contains the local image of F.

For Ω irreducible, Chan–Mok established that each extremal zero set Zero(hα) coincides with Zero(|h′α|^2) for some holomorphic function h′α on C^N that is a C-linear combination of the functions H1,…,HN′ appearing in the expansion of the canonical polynomial ho. The functions H_j for j ≤ N are coordinates, while for ℓ > N they are homogeneous polynomials G_{ℓ−N} of degree at least 2.

The authors explicitly note uncertainty about whether h′α can be chosen to be linear. They subsequently prove a different result (that linear non-degeneracy of F implies sufficient non-degeneracy of the functional system), but this does not resolve the linearity of h′α.