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Analog of the Kazhdan–Lusztig conjecture for quantum Grothendieck rings in general types

Prove the analog of the Kazhdan–Lusztig conjecture for non simply-laced quantum affine algebras by establishing that for every simple module L, the canonical element [L]_t in the quantum Grothendieck ring K_t(C) satisfies π([L]_t) = [L], where π is the natural specialization map to the ordinary Grothendieck ring K(C).

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Background

Nakajima constructed a canonical basis in the quantum Grothendieck ring K_t(C) for simply-laced types, allowing algorithmic computation of q-characters and proving positivity of Kazhdan–Lusztig type polynomials.

For general (non simply-laced) types, positivity has been established, but the full analog of the Kazhdan–Lusztig conjecture—namely that the canonical basis specializes to the simple basis under π—has not been proven.

Resolving this would give a uniform Kazhdan–Lusztig-type character formula for all simple finite-dimensional modules across all types.

References

Although the analog of Kazhdan-Lusztig polynomials are now known to be positive for general types , the analog of the Kazhdan-Lusztig conjecture (that is $\pi([L]_t) = [L]$) is still open in general (in its form formulated for the non simply-laced types in the phD thesis of the speaker ).

— Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models (2510.06437 - Hernandez, 7 Oct 2025) in Section 7 (Quantum Grothendieck rings)