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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Stable maps, Q-operators and category $\mathcal {O}$
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by David Hernandez PDF
Represent. Theory 26 (2022), 179-210 Request permission


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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Additional Information
  • David Hernandez
  • Affiliation: Université Paris Cité and Sorbonne Université, CNRS, IMJ-PRG, IUF, F-75006 Paris, France
  • MR Author ID: 707094
  • Email:
  • Received by editor(s): April 20, 2021
  • Received by editor(s) in revised form: September 17, 2021
  • Published electronically: March 17, 2022
  • Additional Notes: The author was supported by the European Research Council under the European Union’s Framework Programme H2020 with ERC Grant Agreement number 647353 Qaffine
  • © Copyright 2022 American Mathematical Society
  • Journal: Represent. Theory 26 (2022), 179-210
  • MSC (2020): Primary 17B37, 17B10, 82B23
  • DOI:
  • MathSciNet review: 4396040