The center of the affine Hecke algebra

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

Last update: 26 June 2012

The center of H

The center of the affine Hecke algebra is the ring Z(H˜) = 𝕂[P]W = { f𝕂[P] | wf=f for all wW } of symmetric functions in 𝕂[P].

If z 𝕂[P]W then by the fourth relation in (???), Tiz = (siz)Ti + (q-q-1) ( 1-x-αi )-1 (z-siz) = zTi+0, for 1in, and by the third relation in (???), zxλ = xλz, for all λP. Thus z commutes with all the generators of H˜ and so zZ(H˜).

Assume z = λP wW cλ,w xλTw Z(H˜). Let mW be maximal in Bruhat order subject to cγ,m 0 for some γP. If m1 there exists a dominant μP such that c γ+μ-mμ,m = 0 (otherwise c γ+μ-mμ,m 0 for every dominant μP, which is impossible since z is a finite linear combination of xλTw ). Since zZ(H˜) we have z = x-μzxμ = λP wW cλ,w xλ-μ Twxμ. Repeated use of the fourth relation in (???) yields Twxμ = νP vW dν,v xνTv where dν,v are constants such that dwμ,w = 1,  dν,w = 0 for νwμ, and dν,v = 0 unless vw. So z = λP wW cλ,w xλTw = λP wW νP vW cλ,w dν,v xλ-μ+ν Tv and comparing the coefficients of xγTm gives cγ,m = cγ+μ-mμ,m dmμ,m . Since cγ+μ-mμ,m = 0 it follows that cγ,m=0, which is a contradiction. Hence z = λP cλxλ 𝕂[P].

The fourth relation in (???) gives zTi = Tiz = (siz) Ti + (q-q-1)z where z 𝕂[P]. Comparing coefficients of xλ on both sides yields z=0. Hence zTi = (siz)Ti, and therefore z=siz for 1in. So z𝕂 [P]W.

Notes and References

These notes are a retyping, into MathML, of the notes at



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