uniquely determined by the properties stated above. We shall make a definite choice according to [@L1 12.6] and the tables in the appendix of [@L1]. In particular, this means the following. Let us fix a square root of $q$ in $\overline{{{\mathbb{Q}}}}_\ell$. Then it is well-known that the irreducible characters of $G^F$ which occur in the character of the permutation representation of $G^F$ on the cosets of $B^F$ are naturally parametrised by the irreducible characters of $W$; see, e.g., [@L1 8.7]. If $\epsilon\in{{\operatorname{Irr}}}(W)$, we denote the corresponding irreducible character of $G^F$ by $\rho_\epsilon$; clearly, $\rho_\epsilon \in {{\mathfrak{U}}}(G^F)$. Hence, under a bijection ${{\mathfrak{U}}}(G^F)
\leftrightarrow X(W)$ as in \[r11\], the character $\rho_\epsilon$ will correspond to an element $x_{\rho_\epsilon}$. By [@L1 Prop. 12.6], we automatically have $x_\epsilon=x_{\rho_\epsilon}$ except when $\epsilon(1)=
512$ and $G$ is of type $E_7$, or when $\epsilon(1)=4096$ and $G$ is of type $E_8$. In these exceptional cases, ${{\mathfrak{U}}}(G^F)\leftrightarrow X(W)$ can still be chosen such that $x_\epsilon=x_{\rho_\epsilon}$; see the tables for $E_7$, $E_8$ in the appendix of [@L1].

In order to obtain a full uniqueness statement, one has to take into account Harish-Chandra series and further invariants of the characters in ${{\mathfrak{U}}}(G^F)$, namely, the “eigenvalues of Frobenius” as determined in [@L1 11.2]; 