The affine Hecke algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 25 September 2012

The affine Hecke algebra

Let $q$ be an indeterminate and let $𝕂=\mathbb{Z}[q,q^{-1}]\text{.}$ The affine Hecke algebra $\stackrel{\sim }{H}$ is the algebra over $𝕂$ given by generators $T_i,$ $1\le i\le n,$ and $x^{\lambda },$ $\lambda \in P,$ and relations

$$\begin{array}{cc}\begin{array}{cc}\underset{\underset{m_{ij}\text{factors}}{⏟}}{T_i{T}_j{T}_i\dots }=\underset{\underset{m_{ij}\text{factors}}{⏟}}{T_j{T}_i{T}_j\dots },& \text{for all}i\ne j,\\ T_i^2=(q-q^{-1})T_i+1,& \text{for all}1\le i\le n,\\ x^{\lambda }{x}^{\mu }=x^{\mu }{x}^{\lambda }=x^{\lambda +\mu },& \text{for all}\lambda ,\mu \in P,\\ x^{\lambda }{T}_i=T_i{x}^{s_i\lambda }+(q-q^{-1})\frac{x^{\lambda }-x^{s_i\lambda }}{1-x^{-{\alpha }_i}},& \text{for all}1\le i\le n,\lambda \in P\text{.}\end{array}& \text{(1.21)}\end{array}$$

An alternative presentation of $\stackrel{\sim }{H}$ is by the generators $T_w,$ $w\in \stackrel{\sim }{W},$ and relations

$$\begin{array}{cc}{T}_{w_1}{T}_{w_2}=T_{w_1{w}_2},& \text{if}\ell \left(w_1{w}_2\right)=\ell \left(w_1\right)+\ell \left(w_2\right),\\ T_{s_i}{T}_w=(q-q^{-1})T_w+T_{s_{i}w},& \text{if}\ell \left(s_{i}w\right)<\ell \left(w\right)(0\le i\le n)\text{.}\end{array}$$

With notations as in (1.10-1.20) the conversion between the two presentations is given by the relations

$$\begin{array}{cc}\begin{array}{cc}{T}_w=T_{i_1}\dots T_{i_p},& \text{if}w\in W_{\text{aff}}\text{and}w=s_{i_1}\dots s_{i_p}\text{is a reduced word,}\\ T_{g_i}=x^{{\omega }_i}{T}_{w_0{w}_i}^{-1},& \text{for}{g}_i\in \Omega \text{as in (1.19),}\\ x^{\lambda }=T_{t_{\mu }}{T}_{t_{\nu }}^{-1},& \text{if}\lambda =\mu -\nu \text{with}\mu ,\nu \in P^{+},\\ T_{s_0}=T_{s_{\varphi }}{x}^{-\varphi },& \text{where}\varphi \text{is the highest short root of}R\text{.}\end{array}& \text{(1.22)}\end{array}$$

Acknowledgements

The research of A. Ram was partially supported by the National Science Foundation (DMS-0097977), the National Security Agency (MDA904-01-1-0032) and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. The research of K. Nelsen was partially supported by the National Science Foundation (DMS-0097977 and a VIGRE grant) and the National Security Agency (MDA904-01-1-0032).

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