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From Chriss-Ginzburg book "Representation theory and complex geometry", it is written that the relation in the affine Hecke algebra $$ T_s e^{s(\lambda)} - e^{\lambda} T_s = (1-q) \frac{e^{\lambda} - e^{s(\lambda)}}{1 - e^{\alpha}} $$

"clearly" holds for $n\lambda + \lambda'$ if it holds for $\lambda, \lambda'$.

(This algebra has generators $e^{\lambda}T_w$ and certain relations I won't write here. The $e^{\lambda}$ span a subalgebra isomorphic to the group ring of the weight lattice $P$, so they commute. The $T_w$ spans a subalgebra isomorphic to the usual finite Hecke algebra.)

I would appreciate any hint how to deduce the displayed relation (of course it's enough to prove for $n=1$).

I guess the argument is something like "it's multiplicative so we can check it on additive generators of $P$" but the problem is that sum of multiplicative functions is not multiplicative. I also tried to add each side to each other, multiply them or multiply by an exponential without success. It's probably trivial but I can't see it.

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  • $\begingroup$ Try multiplying the formula for $\lambda$ on the right by $e^{s(\lambda’)}$ and the formula for $\lambda’$ on the left by $e^\lambda$, and then add. $\endgroup$ Commented Apr 16, 2021 at 17:46

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More details following Andrew's suggestion : we assume the relation for $\lambda, \lambda'$, and obtain $$ T_s e^{s(\lambda+\lambda')} - e^{\lambda} T_se^{s(\lambda')} = (1-q) \frac{(e^{\lambda} - e^{s(\lambda)})e^{s(\lambda')}}{1 - e^{\alpha}} $$

$$ e^{\lambda}T_s e^{s(\lambda')} - e^{\lambda+\lambda'} T_s = (1-q) \frac{e^{\lambda}(e^{\lambda'} - e^{s(\lambda')})}{1 - e^{\alpha}} $$

Adding the two equations and cancelling certain terms gives the desired relation.

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