By Cédric Bonnafé
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.
Read Online or Download Representations of SL2(Fq) PDF
Best abstract books
Features that are outlined on finite units take place in just about all fields of arithmetic. For greater than eighty years algebras whose universes are such features (so-called functionality algebras), were intensively studied. This booklet provides a extensive advent to the idea of functionality algebras and results in the leading edge of study.
Formes sesquilinéaires et formes quadratiques Les Éléments de mathématique de Nicolas BOURBAKI ont pour objet une présentation rigoureuse, systématique et sans prérequis des mathématiques depuis leurs fondements. Ce neuvième chapitre du Livre d’Algèbre, deuxième Livre du traité, est consacré aux formes quadratiques, symplectiques ou hermitiennes et aux groupes associés.
This ebook deals a scientific presentation of quite a few tools and effects referring to integrable platforms of classical mechanics. The research of integrable structures used to be a massive line of analysis within the final century, yet up till lately just a small variety of examples with or extra levels of freedom have been identified.
This booklet constitutes the refereed complaints of the fifteenth overseas convention on Verification, version Checking and summary Interpretation, VMCAI 2014, held in San Diego, CA, united states, in January 2013. The 25 revised complete papers offered have been conscientiously reviewed and chosen from sixty four submissions. The papers hide a variety of subject matters together with software verification, version checking, summary interpretation and summary domain names, software synthesis, static research, variety platforms, deductive equipment, application certification, debugging ideas, software transformation, optimization, hybrid and cyber-physical structures.
- Ordered Groups
- The Foundations of Analysis: A Straightforward Introduction: Book 1: Logic, Sets and Numbers
- Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory
- A course in abstract algebra
Additional info for Representations of SL2(Fq)
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).
Representations of SL2(Fq) by Cédric Bonnafé