Positivity of an analog of Kazhdan-Lusztig polynomials for finite-dimensional representations of quantum affine algebra

For a complex simple Lie algebra $\mathfrak{g}$, finite-dimensional representations of the associated untwisted quantum affine algebra form an interesting monoidal abelian category, which has been studied from various perspectives. Related to the fundamental problem of determining the characters of irreducible representations in this category, one can consider an analog of Kazhdan-Lusztig polynomials in a purely algebraic way. When $\mathfrak{g}$ is of simply-laced type, the positivity of these polynomials follows from Nakajima's geometric theory of quiver varieties, which is not applicable to non-simply-laced cases. In this talk, we show that the same positivity holds for non-simply-laced type as well by establishing an isomorphism between the quantum Grothendieck ring of non-simply-laced type and that of ''unfolded'' simply-laced type. In addition, we newly find that an analog of Kazhdan-Lusztig conjecture holds for several cases in non-simply-laced type. This is a joint work with David Hernandez, Se-jin Oh, and Hironori Oya.