identity in \[r17\] now reads: $$\rho_{\epsilon}=(-1)^{\dim G} \sum_{x\in X(W)} \zeta_x(A_x:
K_\epsilon^{\mathcal{L}_0}) \chi_x.$$

\[r21\] Let us fix an integer $m{\geqslant}1$. Then $G$ is also defined over ${{\mathbb{F}}}_{q^m}$ and $F^m\colon G\rightarrow G$ is the corresponding Frobenius map. Clearly, $F^m$ acts trivially on $W$ and we have $F^m(t)=
t^{q^m}$ for all $t\in T$. So the whole discussion in Section \[sec1\] can be applied to $F^m$ instead of $F$. As in \[r18\], we write $${{\mathfrak{U}}}(G^{F^m})=\{\rho_x^{(m)} \mid x \in X(W)\}.$$ Again, the unipotent characters of $G^{F^m}$ which occur in the character of the permutation representation of $G^{F^m}$ on the cosets of $B^{F^m}$ are naturally parametrised by ${{\operatorname{Irr}}}(W)$. If $\epsilon\in{{\operatorname{Irr}}}(W)$, we denote the corresponding character of $G^{F^m}$ by $\rho_\epsilon^{(m)}$. As in Remark \[r11a\], the labelling of ${{\mathfrak{U}}}(G^{F^m})$ is arranged such that $$\rho_\epsilon^{(m)}=\rho_{x_\epsilon}^{(m)} \qquad \mbox{for $\epsilon \in 
{{\operatorname{Irr}}}(W)$}.$$ For $y \in X(W)$, the corresponding unipotent almost character of $G^{F^m}$ is given by $$R_y^{(m)}=\sum_{x \in X(W)} \delta_x\{x,y\}\rho_x^{(m)}.$$ Note that $\delta_{\rho_x^{(m)}}=\delta_{\rho_x}=\delta_x$ by [@L1 4.23, 6.20].

\[r22\] Let $x\in X(W)$. Then $\phi_x\colon F^*A_x
\stackrel{\sim}{\rightarrow} A_x$ naturally induces isomorphisms $$F^*(\phi_x)\colon (F^*)^2A_x\stackrel{\sim}{\rightarrow} F^*A_x,\quad
(F^*)^2(\phi_x)\colon (F^*)^3A_x\stackrel{\sim}{\rightarrow} (F^*)^2A_x,
\quad \ldots,$$ which give rise to an isomorphism $$\tilde{\phi}_x^{(m)}:=\phi_x\circ F^*(\phi_x)\circ \ldots \circ 
(F^*)^{m-1}(\phi_x) \colon (F^*)^mA_x\stackrel{\sim}{\rightarrow} A_x.$$ We also have a canonical isomorphism $(F^*)^mA_x\cong (F^m)^* A_x$ which, 