## Two boundary Hecke Algebras and the combinatorics of type $C$

Last updated: 27 January 2015

Abstract

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 ${U}_{q}𝔤$ and the affine braid group of type $C$ on a tensor space $M\otimes N\otimes {V}^{\otimes k}\text{.}$ Special cases provide Schur-Weyl pairings between the affine Hecke algebra of type ${C}_{k}$ and the quantum group of type ${𝔤𝔩}_{n}$ and result in natural labelings of many representations of the affine Hecke algebra 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).

AMS 2010 subject classifications: 20C08 (17B10, 17B37, 05E10)

## Notes and References

This is an excerpt from the paper Two boundary Hecke Algebras and the combinatorics of type $C$ Zajj Daugherty (Department of Mathematics, The City College of New York, NAC 8/133, Convent Ave at 138th Street, New York, NY 10031) and Arun Ram (Department of Mathematics and Statistics, University of Melbourne, Parkville VIC 3010, Australia).