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Quantum groups and flag varieties. (English) Zbl 0818.17018
Sally, Paul J. jun. (ed.) et al., Mathematical aspects of conformal and topological field theories and quantum groups. AMS-IMS-SIAM summer research conference, June 13-19, 1992, Mount Holyoke College, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 175, 101-130 (1994).
The authors extend the classical Schur-Weyl duality to the quantum affine case. In other words, they first “quantize” the setup by replacing the general linear group $$\text{GL}_ n$$ by the quantized universal enveloping algebra of $${\mathfrak {gl}}_ n$$ and the symmetric group $$S_ d$$ by the Hecke algebra of type $$A_ d$$. Then they “affinize” by replacing the field of complex numbers by the $$p$$-adic field.
The approach used is geometric. In fact, the polynomial tensor representations which play the role of the tensor powers of the natural representation for $$\text{GL}_ n$$ in the classical setup is produced by three different geometric constructions. The first is based on equivariant $$K$$-theory, the two others involve the affine flag variety over finite fields.
For the entire collection see [Zbl 0801.00049].

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 14M17 Homogeneous spaces and generalizations