P.S. David's insightful perspective prompts me to add a few remarks about the fumbling I did in the early 1970s from a completely different direction based on inadequate algebraic methods. (At the time I knew little about Lie group representations and less about modern algebraic geometry.) For finite groups of Lie type, it was already a problem to understand the two large series of (mostly irreducible) characters of $SL_2(\mathbb{F}_q)$, having degrees $q+1, q-1$: "principal" and "discrete" series. The principal series is easily constructed by induction from a Borel subgroup, but the origin of the discrete series is mysterious.
After reduction mod $p$ of these characters (with $q$ a power of $p$), it was obvious that both series exhibited roughly the same behavior. Moreover, this paralleled closely the behavior of the "baby" Verma modules for the Lie algebra of the associated algebraic group (using for powers of $p$ the higher Frobenius kernels). In this example all such modules have dimension $q$, with highest weights of composition factors related by "linkage" under an affine Weyl group relative to $p$. I became convinced in general that what I termed "deformation" of linkage classes made all series of finite group characters look essentially the "same"; by 1980 Jantzen had made this formalism precise. But meanwhile Deligne-Lusztig had found a more profound intrinsic way to understand the finite group characters. For me the main moral is the incredible unity found throughout Lie theory.
