Ebook: Affine Flag Varieties and Quantum Symmetric Pairs

The quantum groups of finite and affine type $A$ admit geometric realizations in terms of partial flag varieties of finite and affine type $A$. Recently, the quantum group associated to partial flag varieties of finite type $B/C$ is shown to be a coideal subalgebra of the quantum group of finite type $A$. In this paper the authors study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type $C$. The authors show that the quantum groups arising from Lusztig algebras and Schur algebras via stabilization procedures are (idempotented) coideal subalgebras of quantum groups of affine $mathfrak{sl}$ and $mathfrak{gl}$ types, respectively. In this way, the authors provide geometric realizations of eight quantum symmetric pairs of affine types. The authors construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine $mathfrak{sl}$ type, the authors establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, the authors obtain a new and geometric construction of the idempotented quantum affine $mathfrak{gl}$ and its canonical basis.