## The center of the affine Hecke algebra

Last update: 26 June 2012

## The center of $\stackrel{\sim }{H}$

The center of the affine Hecke algebra is the ring $Z(H˜) = 𝕂[P]W = { f∈𝕂[P] | wf=f for all w∈W }$ of symmetric functions in $𝕂\left[P\right].$

 Proof. If $z\in 𝕂{\left[P\right]}^{W}$ then by the fourth relation in (???), ${T}_{i}z=\left({s}_{i}z\right){T}_{i}+\left(q-{q}^{-1}\right){\left(1-{x}^{-{\alpha }_{i}}\right)}^{-1}\left(z-{s}_{i}z\right)=z{T}_{i}+0,$ for $1\le i\le n,$ and by the third relation in (???), $z{x}^{\lambda }={x}^{\lambda }z,$ for all $\lambda \in P.$ Thus $z$ commutes with all the generators of $\stackrel{˜}{H}$ and so $z\in Z\left(\stackrel{˜}{H}\right).$ Assume $z = ∑ λ∈P w∈W cλ,w xλTw ∈ Z(H˜).$ Let $m\in W$ be maximal in Bruhat order subject to ${c}_{\gamma ,m}\ne 0$ for some $\gamma \in P.$ If $m\ne 1$ there exists a dominant $\mu \in P$ such that ${c}_{\gamma +\mu -m\mu ,m}=0$ (otherwise ${c}_{\gamma +\mu -m\mu ,m}\ne 0$ for every dominant $\mu \in P,$ which is impossible since $z$ is a finite linear combination of ${x}^{\lambda }{T}_{w}$). Since $z\in Z\left(\stackrel{˜}{H}\right)$ we have $z = x-μzxμ = ∑ λ∈P w∈W cλ,w xλ-μ Twxμ.$ Repeated use of the fourth relation in (???) yields $Twxμ = ∑ ν∈P v∈W dν,v xνTv$ where ${d}_{\nu ,v}$ are constants such that for $\nu \ne w\mu ,$ and ${d}_{\nu ,v}=0$ unless $v\le w.$ So $z = ∑ λ∈P w∈W cλ,w xλTw = ∑ λ∈P w∈W ∑ ν∈P v∈W cλ,w dν,v xλ-μ+ν Tv$ and comparing the coefficients of ${x}^{\gamma }{T}_{m}$ gives ${c}_{\gamma ,m}={c}_{\gamma +\mu -m\mu ,m}{d}_{m\mu ,m}.$ Since ${c}_{\gamma +\mu -m\mu ,m}=0$ it follows that ${c}_{\gamma ,m}=0,$ which is a contradiction. Hence $z=\sum _{\lambda \in P}{c}_{\lambda }{x}^{\lambda }\in 𝕂\left[P\right].$ The fourth relation in (???) gives $zTi = Tiz = (siz) Ti + (q-q-1)z′$ where $z\prime \in 𝕂\left[P\right].$ Comparing coefficients of ${x}^{\lambda }$ on both sides yields $z\prime =0.$ Hence $z{T}_{i}=\left({s}_{i}z\right){T}_{i},$ and therefore $z={s}_{i}z$ for $1\le i\le n.$ So $z\in 𝕂{\left[P\right]}^{W}.$ $\square$

## Notes and References

These notes are a retyping, into MathML, of the notes at http://researchers.ms.unimelb.edu.au/~aram@unimelb/Notes2005/cntraffhke7.18.05.pdf

References?