Tokyo Journal of Mathematics

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

Emmanuel LETELLIER

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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.

Article information

Source
Tokyo J. Math., Volume 28, Number 1 (2005), 265-282.

Dates
First available in Project Euclid: 5 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.tjm/1244208292

Digital Object Identifier
doi:10.3836/tjm/1244208292

Mathematical Reviews number (MathSciNet)
MR2149636

Zentralblatt MATH identifier
1105.20035

Citation

LETELLIER, Emmanuel. Deligne-Lusztig Induction for Invariant Functions on Finite Lie Algebras of Chevalley's Type. Tokyo J. Math. 28 (2005), no. 1, 265--282. doi:10.3836/tjm/1244208292. https://projecteuclid.org/euclid.tjm/1244208292


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