Geometric Conjugacy and Deligne–Lusztig Theory

Setting

Let $G$ be a connected reductive group over $\mathbb{F}_q$ with Frobenius endomorphism $F : G \to G$. Write $G^F = G(\mathbb{F}_q)$ for the finite group of rational points. Let $G^*$ be the Langlands dual (or Deligne–Lusztig dual) group over $\mathbb{F}_q$ with dual Frobenius $F^*$.

Pairs $(T, \theta)$

A character datum is a pair $(T, \theta)$ where:

• $T \subset G$ is an $F$-stable maximal torus.
• $\theta : T^F \to \mathbb{C}^\times$ is a character of the finite group $T^F = T(\mathbb{F}_q)$.

Each such pair gives a Deligne–Lusztig virtual character $R_T^\theta \in \mathrm{class\ functions}(G^F)$.

Geometric conjugacy

Two pairs $(T_1, \theta_1)$ and $(T_2, \theta_2)$ are geometrically conjugate if there exists $g \in G(\bar{\mathbb{F}}_q)$ such that $g T_1 g^{-1} = T_2$ and $\theta_2 = \theta_1 \circ \mathrm{Ad}(g^{-1})$ (as characters on $T_2^F$).

This is coarser than $G^F$-conjugacy of pairs: it uses elements from $G(\bar{\mathbb{F}}_q)$, not just $G(\mathbb{F}_q)$.

Parametrisation via $G^*$

The geometric conjugacy classes of pairs $(T, \theta)$ are in bijection with $F^*$-stable semisimple conjugacy classes $[s]$ in $G^*$: $$\{(T, \theta)\}/\text{geom. conj.} \xleftrightarrow{\;\sim\;} \{[s] : s \in G^*, F^*(s) \sim_{\mathrm{conj}} s\}/G^*(\bar{\mathbb{F}}_q).$$

For $G = \mathrm{GL}_n$, this bijects with conjugacy classes of semisimple elements in $\mathrm{GL}_n(\bar{\mathbb{F}}_q)$.

Lusztig series

The Lusztig series $\mathcal{E}(G^F, [s])$ is the set of irreducible characters of $G^F$ that appear as constituents of some $R_T^\theta$ with $(T, \theta)$ geometrically conjugate to the class corresponding to $[s]$.

Lusztig's theorem: The sets $\mathcal{E}(G^F, [s])$ (for $[s]$ ranging over semisimple classes in $G^*$) partition the irreducible characters of $G^F$: $$\mathrm{Irr}(G^F) = \bigsqcup_{[s] \in G^{*F^*}_{\mathrm{ss}}/\sim} \mathcal{E}(G^F, [s]).$$

The series $\mathcal{E}(G^F, [1])$ (for $s = 1$) consists of the unipotent characters and is the most important; its classification reduces to the case $G = G^*$ and involves the theory of character sheaves.

Connection to the Langlands programme

Geometric conjugacy classes of $(T, \theta)$ are the finite-field analogue of unramified Langlands parameters: a semisimple element $s \in G^*$ gives, via the local Langlands correspondence for unramified representations, a Frobenius-semisimple Weil-Deligne parameter $\varphi_s$.