TITLE: Motivation behind the construction of Deligne and Lusztig
QUESTION [20 upvotes]: If $G$ is a connected reductive group over a finite field $\mathbb{F}_q$ and $T$ is a maximal torus in $G$, the famous construction of Deligne and Lusztig (Annals of Math, 1976) associates representations of $G(\mathbb{F}_q)$ to $1$-dimensional representations of $T(\mathbb{F}_q)$. These representations come from the cohomology of the Deligne-Lusztig variety associated to $G$ and $T$, which admits commuting actions of the groups $G(\mathbb{F}_q)$ and $T(\mathbb{F}_q)$. According to various remarks in the Deligne-Lusztig article, two of their sources of motivation were as follows:
1) The conjecture of Macdonald that a construction of this sort should exist.
2) The example of the Drinfeld curve: if $G=SL_2$ and $T$ is the unique (up to conjugacy) non-split maximal torus in $G$, then $T(\mathbb{F}_q)$ can be identified with the kernel of the norm homomorphism
$$\mathbb{F}_{q^2}^\times\to\mathbb{F}_q^\times$$
and $G(\mathbb{F}_q)$ and $T(\mathbb{F}_q)$ both act naturally on the curve $X$ given by the equation $x^qy-xy^q=1$ in the affine plane over $\overline{\mathbb{F}}_q$. (The group $T(\mathbb{F}_q)$ acts by dilations.) The (first) $\ell$-adic cohomology of $X$ realizes all cuspidal irreducible representations of $SL_2(\mathbb{F}_q)$.
In their 1976 paper Deligne and Lusztig give two different constructions of (what later became known as) Deligne-Lusztig varieties (and proved that they are equivalent). The Drinfeld curve turns out to be a special case. However, to me it seems like the jump from the example of the Drinfeld curve (and MacDonald's conjecture) to either of the two general constructions of Deligne-Lusztig is absolutely giant. I was wondering if someone has some additional insight into how the construction was invented.

REPLY [24 votes]: Here is a metaphor which is probably well-known. (I'm repeating what Nick is saying with mild variation.)
When one considers representations of $SL_2(\mathbb{R})$ there are two fundamental homogenous spaces: $\mathbb{P}^1(\mathbb{R})$ and the upper half-plane $\mathbb{H}$. Sections of line bundles ($L^2$ or holomorphic) on these two spaces realize all the admissible representations of $SL_2(\mathbb{R})$.
One can look at this slightly differently: one considers the fundamental homogenous space for the complexified group i.e. $\mathbb{P}^1(\mathbb{C})$ and one decomposes into its rational points $\mathbb{P}^1(\mathbb{R})$ and its complement (the upper and lower hemi-spheres). These two spaces are just (complex conjugate) incarnations of $\mathbb{H}$.
Now do the same thing for $SL_2(\mathbb{F}_q)$: one has the natural action of $SL_2(\mathbb{F_q})$ on $\mathbb{P}^1$ and one can consider the rational points $\mathbb{P}^1(\mathbb{F}_q)) = SL_2(\mathbb{F}_q)/B$ and the complement $X := \mathbb{P}^1 \setminus \{ 0, 1, \dots, p-1 \}$ (an affine algebraic curve). Pursuing the above analogy we might be tempted to call $X$ the "upper half plane" in this context.
If one is optimistic then one might hope that global sections / cohomology of local systems on $X$ realise interesting representations of $SL_2(\mathbb{F}_q)$, which is indeed the case. In this picture the Drinfeld curve emerges as a $T(\mathbb{F}_q)$ (your notation) cover of $X$ which affords a family of interesting local systems via direct image. (Indexed by the characters of $T(\mathbb{F}_q)$.)
In this point of view (local systems on $X$, rather than isotypic components in the cohomology of a cover) the Drinfeld curve loses some of its significance.
Note that there are two elements of the Weyl group: 1 and $s$. We may see $\mathbb{P}^1(\mathbb{F}_q)$ and $X$ as points in relative position $1$ (i.e. equal) and $s$ (i.e. not equal) with their image under Frobenius. Now it is perhaps clearer how to generalise.
(A side remark to be taken with a grain of salt: someone told me once that Drinfeld had the real case in mind when defining Drinfeld space, which is the analogue for p-adic groups. The finite field case was a useful testing ground.)