The center of the affine Hecke algebra

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

Last update: 26 June 2012

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

The center of the affine Hecke algebra is the ring $$Z\left(\tilde{H}\right)=𝕂{\left[P\right]}^W=\{f\in 𝕂\left[P\right]|wf=ffor\; allw\in W\}$$ of symmetric functions in $𝕂\left[P\right].$

		Proof.
	If $z\in 𝕂{\left[P\right]}^W$ then by the fourth relation in (???), $T_{i}z=\left(s_{i}z\right)T_i+(q-q^{-1}){(1-x^{-{\alpha }_i})}^{-1}(z-s_{i}z)=z{T}_i+0,$ 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 $\tilde{H}$ and so $z\in Z\left(\tilde{H}\right).$ Assume $$z=\sum _{\genfrac{}{}{0ex}{}{\lambda \in P}{w\in W}}{c}_{\lambda ,w}{x}^{\lambda }{T}_w\in Z\left(\tilde{H}\right).$$ 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 $z\in Z\left(\tilde{H}\right)$ we have $$z=x^{-\mu }z{x}^{\mu }=\sum _{\genfrac{}{}{0ex}{}{\lambda \in P}{w\in W}}{c}_{\lambda ,w}{x}^{\lambda -\mu }{T}_w{x}^{\mu }.$$ Repeated use of the fourth relation in (???) yields $$T_w{x}^{\mu }=\sum _{\genfrac{}{}{0ex}{}{\nu \in P}{v\in W}}{d}_{\nu ,v}{x}^{\nu }{T}_v$$ where $d_{\nu ,v}$ are constants such that $d_{w\mu ,w}=1, d_{\nu ,w}=0$ for $\nu \ne w\mu ,$ and $d_{\nu ,v}=0$ unless $v\le w.$ So $$z=\sum _{\genfrac{}{}{0ex}{}{\lambda \in P}{w\in W}}{c}_{\lambda ,w}{x}^{\lambda }{T}_w=\sum _{\genfrac{}{}{0ex}{}{\lambda \in P}{w\in W}}\sum _{\genfrac{}{}{0ex}{}{\nu \in P}{v\in W}}{c}_{\lambda ,w}{d}_{\nu ,v}{x}^{\lambda -\mu +\nu }{T}_v$$ 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 $$z{T}_i=T_{i}z=\left(s_{i}z\right)T_i+(q-q^{-1})z\prime$$ 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

References?

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