# Canonical bases arising from quantum symmetric pairs

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## Abstract

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## Notes

### Acknowledgements

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.

## References

- 1.Araki, S.: On root systems and an infinitesimal classification of irreducible symmetric spaces. J. Math. Osaka City Univ.
**13**, 1–34 (1962)MathSciNetMATHGoogle Scholar - 2.Bao, H.: Kazhdan–Lusztig theory of super type \(D\) and quantum symmetric pairs. Represent. Theory
**21**, 247–276 (2017). arXiv:1603.05105 MathSciNetCrossRefMATHGoogle Scholar - 3.Beilinson, A., Lusztig, G., MacPherson, R.: A geometric setting for the quantum deformation of \(GL_n\). Duke Math. J.
**61**, 655–677 (1990)MathSciNetCrossRefMATHGoogle Scholar - 4.Balagovic, M., Kolb, S.: The bar involution for quantum symmetric pairs. Represent. Theory
**19**, 186–210 (2015)MathSciNetCrossRefMATHGoogle Scholar - 5.Balagovic, M., Kolb, S.: Universal \(K\)-matrix for quantum symmetric pairs. J. Reine Angew. Math. (2016). https://doi.org/10.1515/crelle-2016-0012 MATHGoogle Scholar
- 6.Bao, H., Kujawa, J., Li, Y., Wang, W.: Geometric Schur duality of classical type. Transform. Groups. arXiv:1404.4000v3
**(to appear)** - 7.Bao, H., Wang, W.: A new approach to Kazhdan–Lusztig theory of type \(B\) via quantum symmetric pairs, Astérisque. arXiv:1310.0103
**(to appear)** - 8.Bao, H., Wang, W.: Canonical bases in tensor products revisited. Am. J. Math.
**138**, 1731–1738 (2016)MathSciNetCrossRefMATHGoogle Scholar - 9.Ehrig, M., Stroppel, C.: Nazarov–Wenzl algebras, coideal subalgebras and categorified skew Howe duality. arXiv:1310.1972
- 10.Fan, Z., Li, Y.: Positivity of canonical basis under comultiplication. arXiv:1511.02434
- 11.Jantzen, J.C.: Lectures on Quantum Groups, Graduate Studies in Mathematics, vol. 6. American Mathematical Society, Providence (1996)Google Scholar
- 12.Kashiwara, M.: On crystal bases of the \(Q\)-analogue of universal enveloping algebras. Duke Math. J.
**63**, 456–516 (1991)MathSciNetCrossRefMATHGoogle Scholar - 13.Kashiwara, M.: Crystal bases of modified quantized enveloping algebra. Duke Math. J.
**73**, 383–413 (1994)MathSciNetCrossRefMATHGoogle Scholar - 14.Kolb, S.: Quantum symmetric Kac–Moody pairs. Adv. Math.
**267**, 395–469 (2014)MathSciNetCrossRefMATHGoogle Scholar - 15.Kolb, S., Pellegrini, J.: Braid group actions on coideal subalgebras of quantized enveloping algebras. J. Algebra
**336**, 395–416 (2011)MathSciNetCrossRefMATHGoogle Scholar - 16.Letzter, G.: Symmetric pairs for quantized enveloping algebras. J. Algebra
**220**, 729–767 (1999)MathSciNetCrossRefMATHGoogle Scholar - 17.Letzter, G.: Coideal subalgebras and quantum symmetric pairs. In: Montgomery, S., Schneider, H.-J. (eds.) New directions in Hopf algebras (Cambridge), vol. 43, pp. 117–166. MSRI Publications, Cambridge University Press, Cambridge (2002)Google Scholar
- 18.Letzter, G.: Quantum symmetric pairs and their zonal spherical functions. Transform. Groups
**8**, 261–292 (2003)MathSciNetCrossRefMATHGoogle Scholar - 19.Letzter, G.: Quantum zonal spherical functions and Macdonald polynomials. Adv. Math.
**189**, 88–147 (2004)MathSciNetCrossRefMATHGoogle Scholar - 20.Lusztig, G.: Canonical bases arising from quantized enveloping algebras. J. Am. Math. Soc.
**3**, 447–498 (1990)MathSciNetCrossRefMATHGoogle Scholar - 21.Lusztig, G.: Quivers, perverse sheaves, and quantized enveloping algebras. J. Am. Math. Soc.
**4**, 365–421 (1991)MathSciNetCrossRefMATHGoogle Scholar - 22.Lusztig, G.: Canonical bases in tensor products. Proc. Natl. Acad. Sci.
**89**, 8177–8179 (1992)MathSciNetCrossRefMATHGoogle Scholar - 23.Lusztig, G.: Introduction to Quantum Groups, Modern Birkhäuser Classics, Reprint of the 1993 Edition, Birkhäuser, Boston (2010)Google Scholar
- 24.Li, Y., Wang, W.: Positivity vs negativity of canonical bases. In: Proceedings for Lusztig’s 70th Birthday Conference, Bulletin of Institute of Mathematics Academia Sinica (N.S.), vol. 13, pp. 143–198 (2018). arXiv:1501.00688
- 25.Onishchik, A., Vinberg, E.: Lie groups and algebraic groups. Translated from the Russian by D.A. Leites. Springer Series in Soviet Mathematics. Springer, Berlin (1990)Google Scholar