Two boundary Hecke Algebras and combinatorics of type C

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 $C_k$ as the group of braids on $k$ strands with two poles. Generalizing familiar methods from the one pole (type A) case, this provides commuting actions of the quantum group $U_q\mathfrak{g}$ and the affine braid group of type $C_k$ on a tensor space $M\otimes N \otimes V^{\otimes k}$. Special cases provide Schur-Weyl pairings between the affine Hecke algebra of type $C_k$ and the quantum group of type $\mathfrak{gl}_n$, 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 (essentially by identifying their Langlands parameters).