Affine Quantum Symmetric Pairs: Multiplication Formulas and Canonical Bases

Abstract

One breakthrough in the theory of quantum groups is the construction of the canonical bases for quantum groups by Lusztig and Kashiwara. For type A, there is a geometric construction for (idempotented) quantum group together with a canonical basis due to Beilinson, Lusztig and MacPherson (BLM) using a stabilization procedure on a family of quantum Schur algebras of type A. Two essential ingredients in their work are a multiplication formula and a monomial basis. In this dissertation, we provide a BLM-type construction for affine type C. We realize the affine q-Schur algebras of type C as an endomorphism algebra of a certain permutation module of affine Hecke algebras, and then establish a multiplication formula on the Schur algebra level. We provide a direct construction of monomial bases for Schur algebras, which is also adapted to produce monomial bases for affine type A. Via a BLM-type stabilization on the Schur algebras, we construct an algebra K^c admitting canonical basis. We obtain that (K^a, K^c) forms a quantum symmetric pair in the spirit of Letzter and Kolb, where Ka is a quantum group of affine type A. The affine type C construction above is associated to an involution on Dynkin diagrams of affine type A. For other three types of involutions, we construct similar stabilization algebras admitting compatible canonical bases.