$i$, there is a natural isomorphism $$\varphi_{i,\dot{w}}\colon F^*({^p\!H}^i(\bar{K}_{\dot{w}}^{\mathcal{L}_0}))
\stackrel{\sim}{\rightarrow} {^p\!H}^i(\bar{K}_{\dot{w}}^{\mathcal{L}_0}); 
\qquad \mbox{see \cite[12.2, 13.8]{L2c}}.$$ The advantage of using $\bar{K}_{\dot{w}}^{\mathcal{L}_0}$ instead of $K_w^{\mathcal{L}_0}$ (see \[r13\]) is that $\bar{K}_{\dot{w}}^{\mathcal{L}_0}$ is semisimple (see [@L2c 12.8]). Let us now fix $i$, $w$ and denote $K:={^p\!H}^i
(\bar{K}_{\dot{w}}^{\mathcal{L}_0})$. We also set $\varphi:=\varphi_{i,
\dot{w}} \colon F^*K\stackrel{\sim}{\rightarrow}K$ and recall that, for each $x\in X(W)$, we are given $\phi_x\colon F^*A_x \stackrel{\sim}{\rightarrow}
A_x$. Now, following Lusztig [@L2c 13.8.2], there is a canonical isomorphism $$K\cong \bigoplus_{x\in X(W)} (A_x\otimes V_x)$$ where $V_x$ are finite-dimensional vector spaces over $\overline{{{\mathbb{Q}}}}_\ell$ endowed with linear maps $\psi_x\colon V_x\rightarrow V_x$ such that, under the above direct sum decomposition, the map $\phi_x\otimes \psi_x$ corresponds to the given $\varphi\colon F^*K\stackrel{\sim}{\rightarrow} K$. More precisely, $V_x$ and $\psi_x$ are as follows (cf. [@L2b 10.4] and [@L3 3.5]). We have $V_x=\mbox{Hom}(A_x,K)$ and $$\psi_x(v)=\varphi\circ F^*(v)\circ \phi_x^{-1} \qquad\mbox{for
$v\in V_x$},$$ where $F^*(v)\in \mbox{Hom}(F^*A_x,F^*K)$ is the map induced by $v\colon 
A_x\rightarrow K$.

\[r24\] In the setting of \[r23\], all eigenvalues of $\psi_x\colon V_x\rightarrow V_x$ are equal to $\zeta_x q^{(i-\dim G)/2}$.

More precisely, Lusztig first shows in [@L2c 13.10] that there is a constant $0\neq \xi_x\in\overline{{{\mathbb{Q}}}}_\ell$ (which only depends on $\phi_x$, the choice of a square root of $q$ and the choice of the representatives $\dot{w}$) 