Macdonald's conjecture on regular torus characters of finite groups of Lie type

Let $G/{\mathbb F}_q$ be a reductive group, let $T/{\mathbb F}_q$ be a maximal torus contained in $G/{\mathbb F}_q$, and let $\theta:T({\mathbb F}_q)\to{\mathbb C}^{\times}$ be a character. Write

$$W:=N(T)({\mathbb F}_q)/T({\mathbb F}_q),\qquad \epsilon_G:=(-1)^{r(G)},$$

where $r(G)$ is the dimension of a maximal ${\mathbb F}_q$-split torus in $G$, and define $\epsilon_T$ similarly. The character $\theta$ is regular if ${}^w\theta\ne\theta$ for every $w\in W$ with $w\ne1$.

Macdonald's conjecture. For every such regular character $\theta$, there exists an irreducible representation $\pi_\theta$ of $G({\mathbb F}_q)$ of dimension

$$\frac{|G({\mathbb F}_q)|_{p'}}{|T({\mathbb F}_q)|}$$

whose value on a regular semisimple element $s\in G({\mathbb F}_q)$ is nonzero only if $s$ is conjugate to an element of $T({\mathbb F}_q)$. For $s\in T({\mathbb F}_q)$, the character $\chi_\theta$ of $\pi_\theta$ satisfies

$$\chi_\theta(s)=\epsilon_T\epsilon_G\sum_{w\in W}\theta(s^w),$$

and $\chi_\theta$ at unipotent elements is independent of $\theta$.

This conjecture motivated the Deligne–Lusztig theory, which constructs and studies representations associated with characters of maximal tori. The source gives no indication here that the conjecture is resolved, so its status is recorded as open.