Complete proof of the higher-level Garsia–Haiman–Tesler (GHT) identity

Establish a complete algebraic proof of the higher-level GHT identity V()·P_(α^{(1)},…,α^{(r)}|u_1,…,u_r) = W_(α^{(1)},…,α^{(r)}|u_1,…,u_r), for r ≥ 2, in the level-(r,0) horizontal Fock representation of the quantum toroidal gl(1) algebra, without assuming Conjecture BH. Provide an algebraic interpretation that leads to a full proof of the identity using the operators and vertex constructions defined in the paper (including the framing operator F^⊥ and the higher vertex operators).

Background

The paper extends the classical GHT identity to generalized Macdonald functions at level r via vertex-operator techniques, but the complete proof at r>1 currently depends on the unproven Conjecture BH. The authors obtained only a partial proof contingent on this conjecture.

A full proof independent of Conjecture BH would settle the status of the higher-level GHT identity and clarify the algebraic structure behind it. The authors suggest that studying the action of Miki-transformed generators y_{−k}^− = S(x_{−k}^−) and the known action of a_k may uniquely characterize the relevant object, hinting at a possible route to a complete proof.