Canonical bases arising from quantum symmetric pairs

Abstract

We develop a general theory of canonical bases for quantum symmetric pairs $(\mathbf{U},{\mathbf{U}}^{ı})$ with parameters of arbitrary finite type. We construct new canonical bases for the simple integrable $\mathbf{U}$-modules and their tensor products regarded as ${\mathbf{U}}^{ı}$-modules. We also construct a canonical basis for the modified form of the $ı$quantum group ${\mathbf{U}}^{ı}$. 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.

arXiv commentv1, 76 pages. v2, 62 pages, much shortened appendix, modified introduction and other corrections, to appear in Invent. Math

References in corpus (6)

• Quantum symmetric Kac-Moody pairs
• A new approach to Kazhdan-Lusztig theory of type B via quantum symmetric pairs
• The bar involution for quantum symmetric pairs
• Geometric Schur duality of classical type
• Positivity of Canonical Bases under Comultiplication
• Kazhdan-Lusztig Theory of super type D and quantum symmetric pairs

Cited by in corpus (1)

• Formulae of $ı$-divided powers in ${\mathbf{U}}_q({\mathfrak{sl}}_2)$, II