Deligne-Lusztig Induction for Invariant Functions on Finite Lie Algebras of Chevalley's Type

Abstract

Let $G$ be a connected reductive algebraic group defined over $\mathbb{F}_q$ with Lie algebra $\mathcal{G}$. We define a Deligne-Lusztig induction for the $\overline{\mathbb{Q}}_{\ell}$-valued functions on $\mathcal{G}(\mathbb{F}_q)$ which are invariant under the adjoint action of $G(\mathbb{F}_q)$ on $\mathcal{G}(\mathbb{F}_q)$, by making use of the ``character formula'' where the ``two-variable Green functions'' are defined via a $G$-equivariant homeomorphism $\mathcal{G}_{nil}\rightarrow G_{uni}$. We verify that it satisfies properties analogous to the group case like transitivity, the Mackey formula or the commutation with duality. The interest of a Deligne-Lusztig induction for invariant functions comes from a conjecture on a commutation formula with Fourier transforms which has no counterpart in the group case. In a forthcoming paper, this conjecture will be proved in almost all cases.