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Canonical bases arising from quantum symmetric pairs of Kac–Moody type
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- Journal:
- Compositio Mathematica / Volume 157 / Issue 7 / July 2021
- Published online by Cambridge University Press:
- 22 June 2021, pp. 1507-1537
- Print publication:
- July 2021
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- Article
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For quantum symmetric pairs
$(\textbf {U}, \textbf {U}^\imath )$ of Kac–Moody type, we construct 
$\imath$-canonical bases for the highest weight integrable 
$\textbf U$-modules and their tensor products regarded as 
$\textbf {U}^\imath$-modules, as well as an 
$\imath$-canonical basis for the modified form of the 
$\imath$-quantum group 
$\textbf {U}^\imath$. A key new ingredient is a family of explicit elements called 
$\imath$-divided powers, which are shown to generate the integral form of 
$\dot {\textbf {U}}^\imath$. We prove a conjecture of Balagovic–Kolb, removing a major technical assumption in the theory of quantum symmetric pairs. Even for quantum symmetric pairs of finite type, our new approach simplifies and strengthens the integrality of quasi-
$K$-matrix and the constructions of 
$\imath$-canonical bases, by avoiding a case-by-case rank-one analysis and removing the strong constraints on the parameters in a previous work.
vector valued spherical functions and macdonald–koornwinder polynomials
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- Journal:
- Compositio Mathematica / Volume 141 / Issue 5 / September 2005
- Published online by Cambridge University Press:
- 01 September 2005, pp. 1310-1350
- Print publication:
- September 2005
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- Article
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