By Cédric Bonnafé

ISBN-10: 0857291564

ISBN-13: 9780857291561

ISBN-10: 0857291572

ISBN-13: 9780857291578

Deligne-Lusztig concept goals to review representations of finite reductive teams via geometric equipment, and especially l-adic cohomology. many fantastic texts current, with various objectives and views, this conception within the common environment. This booklet makes a speciality of the smallest non-trivial instance, specifically the gang SL2(Fq), which not just give you the simplicity required for an entire description of the speculation, but additionally the richness wanted for illustrating the main smooth aspects.

The improvement of Deligne-Lusztig concept used to be encouraged through Drinfeld's instance in 1974, and Representations of SL2(Fq) relies upon this instance, and extends it to modular illustration idea. To this finish, the writer uses primary result of l-adic cohomology. with the intention to successfully use this equipment, an exact learn of the geometric homes of the motion of SL2(Fq) at the Drinfeld curve is performed, with specific cognizance to the development of quotients through a variety of finite groups.

At the top of the textual content, a succinct evaluate (without evidence) of Deligne-Lusztig thought is given, in addition to hyperlinks to examples confirmed within the textual content. With the availability of either a gradual advent and several other fresh fabrics (for example, Rouquier's theorem on derived equivalences of geometric nature), this e-book could be of use to graduate and postgraduate scholars, in addition to researchers and teachers with an curiosity in Deligne-Lusztig theory.

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Example text

Proof. The μq+1 F mon -equivariance is evident. 1. The surjectivity of υ is clear. We also have υ (x, y ) = υ (x , y ) ⇐⇒ ∃ u ∈ U, (x , y ) = u · (x, y ). Indeed, if (x, y ) ∈ Y and (x , y ) ∈ Y are such that y = y , then x y which shows that q − x x = y y q − x , y x −x x −x ∈ Fq . Now, if we set a = , then y y 1a x · 01 y This shows (2). Point (3) is immediate. = x y . 3. Quotient by μq+1 The morphism π: Y −→ P1 (F) \ P1 (Fq ) (x, y ) −→ [x : y ] is well-defined and is G × F mon -equivariant morphism of varieties (for the action of G induced by the natural action on P1 (F) and the action of F given by [x; y ] → [x q ; y q ]).

Denote by Vθ±0 the irreducible subrepresentation of Vθ0 with character R± (θ0 ). By Schur’s lemma, F acts on Vθ±0 by multiplication by a scalar ρ± . We would like to calculate ρ± . Firstly, we have YF = ∅ and therefore, by the Lefschetz fixed-point formula, we obtain 0 = q − q ρ1 − But (q − 1)(ρ+ + ρ− ) − Tr(F , ⊕ Hc1 (Y)eθ ). 2. Hence ρ− = −ρ+ . 4) To explicitly calculate ρ+ and ρ− , we will study the action of F 2 . As F 2 stabilises Hc1 (Y)eθ , it follows from Schur’s lemma that F 2 acts on Hc1 (Y)eθ by multiplication by a scalar λθ (in fact, if θ = θ0 , then this follows in fact 2 = ρ 2 ).

2). Therefore g is a homothety: g = λ I2 , with λ ∈ Fq× . Now, if (x, y ) ∈ Y, we have (g , ξ ) · (x, y ) = (x, y ), that is λ ξ = 1. Therefore ξ = λ −1 . On the other hand, det(g ) = ξ q+1 , which implies that λ 2 = ξ q+1 or, in other words, λ q+3 = 1. As λ q−1 = 1, we collude that λ 4 = 1, which finishes the proof. Let Δ = D ∩ (G × μq+1 ) = (−I2 , −1) . 4. The group (G × μq+1 )/Δ acts faithfully on Y. Denote by p1 : G → GL2 (Fq ) and p2 : G → Fq×2 the canonical projections, and i1 : μq+1 → G , ξ → (I2 , ξ ) and i2 : G → G , g → (g , 1).

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Representations of SL2(Fq) by Cédric Bonnafé

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