  In order to tackle the problem of generically determining the character
tables of the finite groups of Lie type $\mathbf{G}(q)$ associated to a
connected reductive group $\mathbf{G}$ over $\overline{\mathbb F}_p$, Lusztig
developed the theory of character sheaves in the 1980s. The subsequent work of
Lusztig and Shoji in principle reduces this problem to specifying certain roots
of unity. The situation is particularly well understood as far as character
values at unipotent elements are concerned. We complete the computation of the
values of unipotent characters at unipotent elements for the groups
$\mathbf{G}(q)$ where $\mathbf{G}$ is the simple group of type $E_8$, by
determining the aforementioned roots of unity.

Representation Theory