3. Deligne-Lusztig Induction

Abstract

From now we assume that $k = \overline{\mathbb{F}}_q$ with q a power of p and that G is defined over $\mathbb{F}_q$. Let $\ell$ denote a prime not equal to p and $\overline{\mathbb{Q}}_{\ell}$ an algebraic closure of the field $\mathbb{Q}_{\ell}$ of $\ell$-adic numbers. In this chapter, we first recall some facts about the $\overline{\mathbb{Q}}_{\ell}$-space $\mathcal{C}(\mathcal{G}^F)$ of all functions $\mathcal{G}^F\rightarrow\overline{\mathbb{Q}}_{\ell}$ which are invariant under the adjoint action of G^F on $\mathcal{G}^F$. We then define, when p is good for G, a Lie algebra version of Deligne-Lusztig induction [DL76], that is for any F-stable Levi subgroup L of G with Lie algebra $\mathcal{L}$, we define a $\overline{\mathbb{Q}}_{\ell}$-linear map $\mathcal{R}_{\mathcal{L}}^{\mathcal{G}}: \mathcal{C}(\mathcal{L}^F)\rightarrow\mathcal{C}(\mathcal{G}^F)$ which satisfies analogous properties to the group case, like transitivity, the Mackey formula and commutation with the duality map. We finally formulate as a conjecture a property which has no counterpart in the group setting, namely that the Deligne-Lusztig induction commutes with Fourier transforms.