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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 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.

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# Motivation behind the construction of Deligne and Lusztig

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.