Generalized HL-modules and cluster algebras

We define a family of finite-dimensional simple [Formula: see text]-modules by their highest [Formula: see text]-weight monomials, call them generalized HL-modules as a generalization of Hernandez–Leclerc modules of type [Formula: see text], and prove that generalized HL-modules are real and prime. We also prove that Hernandez–Leclerc conjectures hold for a family of generalized HL-modules, that is, their equivalence classes are cluster variables in the Grothendieck ring [Formula: see text] of a monoidal subcategory [Formula: see text] of the category of finite-dimensional [Formula: see text]-modules.