{\bf Addendum just above 15.9}\par
Once we know the unipotent characters of $\GF$, we can easily get all
characters, using 13.30. Indeed we can take $(\bGd,F^*)$ to be $(\bG,F)$;
see examples above 13.11.
Moreover the centralizer of a semi-simple element is a Levi  subgroup  by
2.6,  and is isomorphic to a group of block-diagonal matrices.  If $s$ is
rational semi-simple, by 4.3 the action of  $F$  on  $C_\bG(s)$  permutes
blocks  of  equal  size and the smallest power of $F$ which fixes a block
still acts on  that  block  as  a  standard  or  unitary  type  Frobenius
