Is the representation of finite simple groups fully understood?

Question

Is the representation of finite simple groups fully understood? To clarify, I mean have all the simple representations (even finite dimensional) been classified in terms of some classifying set, such as we have for simple Lie algebras/groups.

Answer 1

A term that may fit in the scope of this problem is "generic character table", character tables of a whole family of groups of Lie type.

Example: Generic character table of $SL_2(q)$, $q = 2^f$

Representations	$I$	$U$	$S(a)$	$T(b)$
Trivial	$1$	$1$	$1$	$1$
Steinberg	$q$	$0$	$1$	$-1$
Principal indexed by $k=1 \dots q/2 − 1$	$q+1$	$1$	$\epsilon^{ak}$$+$$\epsilon^{−ak}$	$0$
Discrete indexed by $l=1 \dots q/2$	$q-1$	$-1$	$0$	$−\eta^{bl}$$−$$\eta^{−bl}$

where $ \epsilon = \exp (2πi/(q − 1))$, $η = exp (2πi/(q + 1))$.

The generic character tables of the groups of Lie type $D_5(q)$ and $E_6(q)$ are still unknown, let alone $E_7(q)$ and $E_8(q)$.

And even if we worked out the whole character table, there is still a gap from the character table to representations. To see this, you can try to derive the representations from the character table above.