Parametrization via χ(Z), part (i): sufficiency for membership in O^{sh}_{Z}

Prove that if the highest ℬ-parameter Ψ of a simple module arises from a monomial in χ(Z) as defined in the text, then the corresponding simple module L(Ψ) lies in the truncated shifted category O^{sh}_{Z}.

Background

For a truncation parameter Z=(Z_i(z)) with roots encoded by the exponents u_{i,a}, the product χ(Z)=∏{i,a} χ{i,a^{-1}}^{u_{i,a}} is formed; from any monomial M in χ(Z), one constructs Ψ_M via prescribed substitutions.

This conjectural criterion provides a direct representation-theoretic interpretation of χ(Z) by identifying which simple modules descend to the truncated shifted quantum affine algebra U_{q,Z}^μ(ĝ).

Part (i) asserts a sufficiency statement: if Ψ arises from χ(Z), then L(Ψ) should belong to O^{sh}_{Z}.