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3. Deligne-Lusztig Induction

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1859)

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 GF 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.

Mathematics Subject Classification (2000):

  • 20C33

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Correspondence to Emmanuel Letellier .

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© 2005 Springer-Verlag Berlin/Heidelberg

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Letellier, E. (2005). 3. Deligne-Lusztig Induction. In: Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras. Lecture Notes in Mathematics, vol 1859. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31561-2_3

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