# Hecke algebra relations

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.

• 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. Commented Apr 16, 2021 at 17:46

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