Abstract
The Schur-Jimbo duality is one of the most fundamental topics in representation theory, bridging the irreducible representations of a Hecke algebra with those of a Drinfeld-Jimbo quantum group. Evolving alongside the advancements in the field, the Schur-Jimbo duality has been extended in tandem with the emergence of $\imath$quantum groups, which are a natural generalization of quantum groups arising from the theory of quantum symmetric pairs.
In this dissertation, we construct various $\imath$Schur dualities stemming from quantum symmetric pairs of types AI, AII, and AIII. Particularly, the $\imath$Schur duality of type AIII, accommodating black nodes in its Satake diagram, presents a unified extension of Jimbo-Schur duality and Bao-Wang's quasi-split $\imath$Schur duality.
Moreover, expanding the classic works of Kazhdan-Lusztig and Deodhar, we establish bar involutions and canonical bases on quasi-permutation modules over the type B Hecke algebra, where the bases are parameterized by cosets of (possibly non-parabolic) reflection subgroups of the Weyl group of type B. The quasi-parabolic KL bases on quasi-permutation Hecke modules are shown to match with the $\imath$canonical basis on the tensor space.
Finally, we establish two specific families of quantum supersymmetric pairs, denoted as type AIII and type AI-II, respectively. We elucidate their fundamental properties, including the coideal algebra property and the quantum Iwasawa decomposition, which ensure that the $\imath$quantum supergroups attain the expected sizes. Within the framework of these quantum supersymmetric pairs, we provide super generalizations of the aforementioned dualities.
