Stable maps, Q-operators and category 𝒪

Hernandez, David

doi:10.1090/ert/604

Stable maps, Q-operators and category $\mathcal {O}$
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by David Hernandez

Represent. Theory 26 (2022), 179-210

DOI: https://doi.org/10.1090/ert/604

Published electronically: March 17, 2022

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Abstract:

Motivated by Maulik-Okounkov stable maps associated to quiver varieties, we define and construct algebraic stable maps on tensor products of representations in the category $\mathcal {O}$ of the Borel subalgebra of an arbitrary untwisted quantum affine algebra. Our representation-theoretical construction is based on the study of the action of Cartan-Drinfeld subalgebras. We prove the algebraic stable maps are invertible and depend rationally on the spectral parameter. As an application, we obtain new $R$-matrices in the category $\mathcal {O}$ and we establish that a large family of simple modules, including the prefundamental representations associated to $Q$-operators, generically commute as representations of the Cartan-Drinfeld subalgebra. We also establish categorified $QQ^*$-systems in terms of the $R$-matrices we construct.

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