## The affine Hecke algebra

Last update: 25 September 2012

## The affine Hecke algebra

Let $q$ be an indeterminate and let $𝕂=ℤ\left[q,{q}^{-1}\right]\text{.}$ 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

$TiTjTi… ⏟ mijfactors = TjTiTj… ⏟ mijfactors , for alli≠j, Ti2= (q-q-1)Ti +1, for all 1≤i≤n, xλxμ=xμxλ =xλ+μ, for all λ,μ∈P, xλTi=Ti xsiλ+ (q-q-1) xλ- xsiλ 1-x-αi , for all 1≤i≤n,λ∈P. (1.21)$

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

$Tw1 Tw2= Tw1w2, ifℓ(w1w2) =ℓ(w1)+ ℓ(w2), TsiTw= (q-q-1)Tw+ Tsiw, ifℓ(siw)< ℓ(w) (0≤i≤n).$

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

$Tw=Ti1… Tip, ifw∈Waff andw=si1… sipis a reduced word, Tgi= xωi Tw0wi-1, forgi∈Ω as in (1.19), xλ=Ttμ Ttν-1, ifλ=μ-νwith μ,ν∈P+, Ts0=Tsϕ x-ϕ, whereϕis the highest short root ofR . (1.22)$

## 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).