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In this talk, I will introduce the quantum affine analog of Kazhdan-Lusztig(KL) positivity conjecture suggested by Hernandez.
The conjecture is already proved by Nakajima in a geometric way , when the quantum affine algebra is of simply-laced type.
By establishing isomorphism between their Grothendieck rings for (simply-laced g_1 and non- simply-laced g_2) in a systematic way,
we can propagate the positivity in simply laced type to non- simply laced type. Joining the result of Kashiwara-Kim-myself, we prove further that the (q,t)-character of each simple module of type $B$ is "canonical" $t$-deformtation of its q-character.
This is joint work with Fujita-Hernandez-Oya (arXiv:2101.07489) and Fujita (arXiv:2007.03159).