Affine flag varieties and quantum symmetric pairs

Abstract

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 we study the structures of Schur algebras and Lusztig algebras associated to (four variants of) partial flag varieties of affine type $C$. We 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, we provide geometric realizations of eight quantum symmetric pairs of affine types. We construct monomial and canonical bases of all these quantum (Schur, Lusztig, and coideal) algebras. For the idempotented coideal algebras of affine $\mathfrak{sl}$ type, we establish the positivity properties of the canonical basis with respect to multiplication, comultiplication and a bilinear pairing. In particular, we obtain a new and geometric construction of the idempotented quantum affine $\mathfrak{gl}$ and its canonical basis.

v1. 108 pages. v2. 113 pages. Minor revisions with a list of notations added. Reference updated. To appear in the Memoirs of the AMS. Footnotes added in pages 25, 39, 51, 65 and 74 to fix typos found after publication

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• Quantum Schur duality of affine type C with three parameters
• Cells in affine q-Schur algebras
• Multiplicaton formulas and canonical basis for quantum affine gl_n