Inventiones mathematicae

, Volume 213, Issue 3, pp 1099–1177 | Cite as

Canonical bases arising from quantum symmetric pairs

  • Huanchen Bao
  • Weiqiang WangEmail author


We develop a general theory of canonical bases for quantum symmetric pairs \(({\mathbf{U}}, {\mathbf{U}}^\imath )\) with parameters of arbitrary finite type. We construct new canonical bases for the finite-dimensional simple \({\mathbf{U}}\)-modules and their tensor products regarded as \({\mathbf{U}}^\imath \)-modules. We also construct a canonical basis for the modified form of the \(\imath \)quantum group \({\mathbf{U}}^\imath \). To that end, we establish several new structural results on quantum symmetric pairs, such as bilinear forms, braid group actions, integral forms, Levi subalgebras (of real rank one), and integrality of the intertwiners.

Mathematics Subject Classification

Primary 17B10 



HB is partially supported by an AMS-Simons travel grant, and WW is partially supported by an NSF grant. We thank the following institutions whose support and hospitality helped to greatly facilitate the progress of this project: East China Normal University, Institute for Advanced Study, Institute of Mathematics at Academia Sinica, and Max-Planck Institute for Mathematics. We would like to thank Jeffrey Adams, Xuhua He, Stefan Kolb, and George Lusztig for helpful discussions, comments and their interest. An earlier version on the \(\imath \)-canonical basis construction in the special cases when \(\mathbb {I}_{\bullet }=\emptyset \) (for finite and affine types) was completed in Spring 2015. Balagovic and Kolb’s work (in their goal of showing the universal \(\mathcal {K}\)-matrix provides solutions to the reflection equation) helped to address several foundational issues on QSP raised in our 2013 announcement on the program of canonical basis for general QSP, and we in turn use their results in the current version. We warmly thank them for their valuable contributions. We are grateful to a referee for careful readings and numerous suggestions and corrections.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MarylandCollege ParkUSA
  2. 2.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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