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On a Theorem of Shintani

✍ Scribed by Cédric Bonnafé

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
136 KB
Volume
218
Category
Article
ISSN
0021-8693

No coin nor oath required. For personal study only.

✦ Synopsis

Let be an irreducible character of G s GL ‫ކ‬ invariant under the

automorphism of G induced by the field automorphism ‫ކ‬ d ª ‫ކ‬ d, x ¬ x q , and

let e be a divisor of d. By a theorem of Shintani, there exists an extension of ˜e ² e : to G i whose Shintani descent to G is, up to a sign , an irreducible d e character of G . It is shown in this paper that may always be chosen such that ẽ e s 1. With this particular choice, is the restriction of . Our methods rely on ˜ẽ 1 the work of Digne and Michel on Deligne᎐Lusztig theory for nonconnected reductive groups. ᮊ 1999 Academic Press Ž . d Let GЊ s GL ‫ކ‬ , where ‫ކ‬ is an algebraic closure of a finite field and n where n and d are natural numbers. The symmetric group ᑭ acts on GЊ d by permutations of the components of GЊ. We denote by G the semidirect product G s GЊ i ᑭ . It is a nonconnected reductive group, with neutral d component GЊ. We denote by F : G ª G the natural split Frobenius 0

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