A problem on Kazhdan–Lusztig theorem

Question

I am reading Chriss, Ginzburg's book Representation theory and complex geometry. In theorem 7.2.16 it says that the convolution action of the Steinberg variety $St=\tilde{\mathcal{N}}\times_\mathfrak{g}\tilde{\mathcal{N}}$ on the Springer resolution $\tilde{\mathcal{N}}$ induces an action on $K$-groups, i.e. an action of $K_0^{G\times\mathbb{G}_m}(St)=H_\text{aff}$ on $K_0^{G\times\mathbb{G}_m}(\tilde{\mathcal{N}})=R(T)$. Then the claim is that the element $T_{s_\alpha}\in H_\text{aff}$ acts as $$T_{s_\alpha}:e^\lambda\mapsto\frac{e^\lambda-e^{s_\alpha(\lambda)}}{e^{\alpha}-1}-q\frac{e^\lambda-e^{s_\alpha(\lambda)+\alpha}}{e^{\alpha}-1}$$ where $\alpha$ is a simple root, $\lambda$ is a weight of $T$. It should be true that the subalgebra $R(T)\subset H_\text{aff}$ acts on $R(T)$ by multiplication(This is where I've made a mistake. It should act on $R(T)$ by multiply its inverse).

On the other hand we have the Bernstein relations which tell us that in $H_\text{aff}$ there is an equation $$T_{s_\alpha}e^\lambda=e^{s_\alpha(\lambda)}T_{s_\alpha}+(q-1)\frac{e^\lambda-e^{s_\alpha(\lambda)}}{1-e^{-\alpha}}.$$ But it seems that this two equations are not compatible if we consider the homomorphism $H_\text{aff}\to \operatorname{End}(R(T))$. What is the problem here?

Answer 1

In the construction of $K_0^{G\times\mathbb{G_m}}(St)\cong H_{\mathrm{aff}}$ the class of $\Delta_* L_\lambda$ is mapped to $e^{-\lambda}$, so the action of $e^{\lambda}\in R(T)\subset H_{\mathrm{aff}}$ on $R(T)$ should be multiplication by $e^{-\lambda}$. Then everything will be good.