The affine Hecke algebra

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 25 September 2012

The affine Hecke algebra

Let q be an indeterminate and let 𝕂=[q,q-1]. The affine Hecke algebra H is the algebra over 𝕂 given by generators Ti, 1in, and xλ, λP, and relations

TiTjTi mijfactors = TjTiTj mijfactors , for allij, Ti2= (q-q-1)Ti +1, for all 1in, xλxμ=xμxλ =xλ+μ, for all λ,μP, xλTi=Ti xsiλ+ (q-q-1) xλ- xsiλ 1-x-αi , for all 1in,λP. (1.21)

An alternative presentation of H is by the generators Tw, wW, and relations

Tw1 Tw2= Tw1w2, if(w1w2) =(w1)+ (w2), TsiTw= (q-q-1)Tw+ Tsiw, if(siw)< (w) (0in).

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

Tw=Ti1 Tip, ifwWaff andw=si1 sipis a reduced word, Tgi= xωi Tw0wi-1, forgiΩ as in (1.19), xλ=Ttμ Ttν-1, ifλ=μ-νwith μ,νP+, Ts0=Tsϕ x-ϕ, whereϕis the highest short root ofR . (1.22)


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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