  After establishing a geometric Schur-Weyl duality in a general setting, we
recall this duality in type A in the finite and affine case. We extend the
duality in the affine case to positive parts of the affine algebras. The
positive parts have nice ideals coming from geometry, allowing duality for
quotients. Some of the quotients of the positive affine Hecke algebra are then
identified to some cyclotomic Hecke algebras and the geometric setting allows
the construction of canonical bases.
