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After establishing a geometric Schur-Weyl duality in a general setting, werecall this duality in type A in the finite and affine case. We extend theduality in the affine case to positive parts of the affine algebras. Thepositive parts have nice ideals coming from geometry, allowing duality forquotients. Some of the quotients of the positive affine Hecke algebra are thenidentified to some cyclotomic Hecke algebras and the geometric setting allowsthe construction of canonical bases.

A geometric Schur-Weyl duality for quotients of affine Hecke algebras