Two boundary Hecke algebras and combinatorics of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>C</mml:mi> </mml:math>

This paper gives a Schur–Weyl duality approach to the representation theory of the affine Hecke algebras of type C with unequal parameters. The first step is to realize the affine braid group of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> as the group of braids on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>k</mml:mi> </mml:math> strands with two poles. Generalizing familiar methods from the one pole (type A) case, this provides commuting actions of the quantum group <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mi>𝔤</mml:mi> </mml:mrow> </mml:math> and the affine braid group of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> on a tensor space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>⊗</mml:mo> <mml:mi>N</mml:mi> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mi>V</mml:mi> <mml:mrow> <mml:mo>⊗</mml:mo> <mml:mi>k</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . Special cases provide Schur–Weyl pairings between the affine Hecke algebra of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> and the quantum group of type <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>𝔤𝔩</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> , resulting in natural labelings of many representations of the affine Hecke algebras of type C by partitions. Following an analysis of the structure of weights of affine Hecke algebra representations (extending the one parameter case to the three parameter case necessary for affine Hecke algebras of type C), we provide an explicit identification of the affine Hecke algebra representations that appear in tensor space.