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In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by . used these representations to find all representations of all finite simple groups of Lie type. ==Motivation== Suppose that ''G'' is a reductive group defined over a finite field, with Frobenius map ''F''. Macdonald conjectured that there should be a map from ''general position'' characters of ''F''-stable maximal tori to irreducible representations of ''G''''F'' (the fixed points of ''F''). For general linear groups this was already known by the work of . This was the main result proved by Deligne and Lusztig; they found a virtual representation for all characters of an ''F''-stable maximal torus, which is irreducible (up to sign) when the character is in general position. When the maximal torus is split, these representations were well known and are given by parabolic induction of characters of the torus (extend the character to a Borel subgroup, then induce it up to ''G''). The representations of parabolic induction can be constructed using functions on a space, which can be thought of as elements of a suitable zeroth cohomology group. Deligne and Lusztig's construction is a generalization of parabolic induction to non-split tori using higher cohomology groups. (Parabolic induction can also be done with tori of ''G'' replaced by Levi subgroups of ''G'', and there is a generalization of Deligne–Lusztig theory to this case too.) Drinfeld proved that the discrete series representations of SL2(F''q'') can be found in the ℓ-adic cohomology groups :H1c(''X'', Q''ℓ'') of the affine curve ''X'' defined by :''xyq−yxq'' = 1. The polynomial ''xyq−yxq'' is a determinant used in the construction of the Dickson invariant of the general linear group, and is an invariant of the special linear group. The construction of Deligne and Lusztig is a generalization of this fundamental example to other groups. The affine curve ''X'' is generalized to a ''T''''F'' bundle over a "Deligne–Lusztig variety" where ''T'' is a maximal torus of ''G'', and instead of using just the first cohomology group they use an alternating sum of ℓ-adic cohomology groups with compact support to construct virtual representations. The Deligne-Lusztig construction is formally similar to Weyl's construction of the representations of a compact group from the characters of a maximal torus. The case of compact groups is easier partly because there is only one conjugacy class of maximal tori. The Borel–Weil–Bott construction of representations of algebraic groups using coherent sheaf cohomology is also similar. For real semisimple groups there is an analogue of the construction of Deligne and Lusztig, using Zuckerman functors to construct representations. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Deligne–Lusztig theory」の詳細全文を読む スポンサード リンク
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