Deligne–Lusztig theory

Definition

Let $p$ be a prime, let $q$ be a power of $p$, and let $\overline{\F}_q$ be an algebraic closure of the finite field $\F_q$. Let $\mathbf{G}$ be a connected reductive group over $\overline{\F}_q$ with a Frobenius endomorphism $F$ for an $\F_q$-structure. Let $G = \mathbf{G}^F$ be the finite group of fixed points of $F$. Let $\mathbf{T}$ be an $F$-stable maximal torus of $\mathbf{G}$, and let $T = \mathbf{T}^F$. Let $\mathbf{B}$ be a Borel subgroup containing $\mathbf{T}$, not necessarily $F$-stable, with unipotent radical $\mathbf{U}$. The Deligne–Lusztig variety is $$\tilde{X} = \set{g \in \mathbf{G} : g^{-1} F(g) \in F(\mathbf{U})}.$$ The group $G$ acts on $\tilde{X}$ by left multiplication and $T = \mathbf{T}^F$ acts by right multiplication. Let $\ell \ne p$ be a prime and let $\theta \colon T \to \overline{\Q}_\ell^\times$ be a character. Let $H^i_c(\tilde{X})_\theta$ be the subspace of the $\ell$-adic cohomology with compact support $H^i_c(\tilde{X}, \overline{\Q}_\ell)$ on which $T$ acts through $\theta$. The Deligne–Lusztig virtual character is the function on $G$ $$R_{\mathbf{T}}^\theta(g) = \sum_{i \ge 0} (-1)^i \tr\big(g, H^i_c(\tilde{X})_\theta\big).$$ Deligne–Lusztig theory is the study of the representations of $G$ through the virtual characters $R_{\mathbf{T}}^\theta$.