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Flag manifolds.

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Flag industry --- Law and legislation

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Affine flag manifolds are infinite dimensional versions of familiar objects such as Graßmann varieties. The book features lecture notes, survey articles, and research notes - based on workshops held in Berlin, Essen, and Madrid - explaining the significance of these and related objects (such as double affine Hecke algebras and affine Springer fibers) in representation theory (e.g., the theory of symmetric polynomials), arithmetic geometry (e.g., the fundamental lemma in the Langlands program), and algebraic geometry (e.g., affine flag manifolds as parameter spaces for principal bundles). Novel aspects of the theory of principal bundles on algebraic varieties are also studied in the book.

Geometry. --- Mathematics. --- Flag manifolds --- Mathematics --- Physical Sciences & Mathematics --- Geometry --- Flag manifolds. --- Fiber spaces (Mathematics) --- Fibre spaces (Mathematics) --- Flag varieties (Mathematics) --- Manifolds, Flag --- Varieties, Flag (Mathematics) --- Algebraic geometry. --- Algebraic Geometry. --- Algebraic topology --- Algebraic varieties --- Geometry, algebraic. --- Algebraic geometry

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Air quality indexes --- Environmental health --- Air quality --- Standards --- Air Quality Flag Program (U.S.) --- United States.

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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.

Flag manifolds. --- Affine algebraic groups. --- Quantum groups. --- Schur complement. --- Kazhdan-Lusztig polynomials. --- Algebra, Homological.