Motivation behind the construction of Deligne and Lusztig

Question

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.

Answer 1

It's not easy to explain the motivation without being one of the authors, but in fact Lusztig has provided some helpful perspective on the writing of his joint paper with Deligne (1976) and his earlier related paper (1974) in Ann. of Math. Studies 81. On his homepage at MIT you can find an intimidating list of all his papers here, along with detailed comments on some of them here. See in particular numbers 17 and 22. Even though his comments are fairly short, they do bring out the transition from the earlier ideas of Macdonald and Springer to the specific construction of Deligne-Lusztig varieties. Some of the personal contacts and influences are impossible to trace, but a basic motivation was the construction of explicit representations of the finite groups of Lie type which would realize the elusive "cuspidal" or "discrete series" characters. In his 1955 paper on finite general linear groups, Green was able to deal with the characters inductively in a combinatorial spirit, but for other Lie types the story gets more complicated and requires a more sophisticated approach.

There were of course some reviews of the two papers I've mentioned, along with a nice technical survey by Serre in the 1975-76 Bourbaki seminar. But it's hard to extract from the literature as much insight as you can get from Lusztig's own comments. In particular, I think he makes it clear that there was no single moment of illumination based on the rank 1 case, but rather a coming together of a number of ways of thought that had already become influential in algebraic geometry and representation theory (illustrated by Springer's work on representations of Weyl groups in the early 1970s). Lusztig himself started out in algebraic topology but his collaboration with Roger Carter in Warwick got him involved in some of the problems of representation theory for algebraic groups and finite groups of Lie type. Having said all this, it must be added that it takes some rather brilliant people to come up with the right approach to such a stubborn problem in finite group theory.

Answer 2

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.)