In this paper we construct a family of irreducible representations of a Chevalley group over a finite ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> …
In this paper we construct a family of irreducible representations of a Chevalley group over a finite ring <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R"> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding="application/x-tex">R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of truncated power series over a field <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper F Subscript q"> <mml:semantics> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">\mathbf F_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This is done by a cohomological method extending that of Deligne and the author in the case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R equals bold upper F Subscript q"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="bold">F</mml:mi> </mml:mrow> <mml:mi>q</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">R=\mathbf F_q</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
We discuss a cohomological construction of representations of a reductive group over the ring of power series over a finite field modulo the r-th power of the maximal ideal. The …
We discuss a cohomological construction of representations of a reductive group over the ring of power series over a finite field modulo the r-th power of the maximal ideal. The case r=1 goes back to Deligne and the author. The case where r is greater than 1 was given without proof in the author's work in 1977. Here we supply the proofs and give some further results.
We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility properties …
We discuss progress towards the classification of irreducible admissible representations of reductive groups over non-archimedean local fields and the local Langlands correspondence. We also state some (partly conjectural) compatibility properties of the refined local Langlands correspondence.
conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field F , in terms of local Langlands parameters.In …
conjectured an explicit expression for the Plancherel density of the group of points of a reductive group defined over a local field F , in terms of local Langlands parameters.In these lectures we shall present a proof of these conjectures for Lusztig's class of representations of unipotent reduction if F is p-adic and G is of adjoint type and splits over an unramified extension of F .This is based on the author's paper [Spectral transfer morphisms for unipotent affine Hecke algebras, Selecta Math.(N.S.) 22 (2016), no. 4, 2143-2207].More generally for G connected reductive (still assumed to be split over an unramified extension of F ), we shall show that the requirement of compatibility with the conjectures of Hiraga, Ichino and Ikeda essentially determines the Langlands parameterisation for tempered representations of unipotent reduction.We shall show that there exist parameterisations for which the conjectures of Hiraga, Ichino and Ikeda hold up to rational constant factors.The main technical tool is that of spectral transfer maps between normalised affine Hecke algebras used in op.cit.
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Let $S$ be a reduced, irreducible scheme and $G$ a reductive group scheme over $S$ .A representation of $G$ is, by definition, a pair $(\rho, V)$ of a vector bundle …
Let $S$ be a reduced, irreducible scheme and $G$ a reductive group scheme over $S$ .A representation of $G$ is, by definition, a pair $(\rho, V)$ of a vector bundle $V$ over $S$ and a homomorphism $\rho:G\rightarrow GL(V)$ .If $\overline{\eta}$ is the generic geo- metric point of $S$ , we call $(G, \rho, V)$ an S-form of $(G_{\overline{\eta}}, \rho_{\overline{\eta}}, V_{\overline{\eta}})$ .The purpose of this paper is to describe the S-forms of an irreducible representation of $G_{\overline{\eta}}$ , assuming that $S$ is normal and locally noetherian.As is well known, if $S$ is the prime spectrum of a field, the S-forms of a given representation can be obtained by twisting the split S-form using the Galois cohomology.In the general case, the S-forms of a given representation can be also obtained by twisting the split ones using a non-abelian \'etale co- nomology, which is a natural generalization of the usual Galois cohomology.In contrast with the case where $S$ is the prime spectrum of a field, there are possibly more than one split S-forms.The results of this paper will be applied to a study of prehomogeneous vector spaces.Conventions.Since we refer [5] very often, we shall write [$Exp$ .$X$ , Y.Z. $]$for $[5; Exp.X, Y.Z.\cdots]$ .If we are considering an algebraic variety $V$ over an algebraically close field $K$ , we often identify $V$ with the set of rational points $V(K)$ .If a scheme $X$ is considered as a scheme over another scheme $S$ , we add suffix $S$ and write $X_{S}$ .If $S=SpecA$, we write $X_{A}$ for $X_{SpecA}$ .1. Representations of Chevalley-Demazure group schemes.The purpose of this section is to describe the irreducible representations of a Chevalley-Demazure group scheme.The main result of this section is (1.19).1.1.Let $K$ be an algebraically closed field, $G_{K}$ a (connected) reductive algebraic group over $K,$ $T_{K}$ a maximal torus of $G_{K},$ $B_{K}$ a Borel subgroup of
This book presents a classification of all (complex) irreducible representations of a reductive group with connected centre, over a finite field. To achieve this, the author uses etale intersection cohomology, …
This book presents a classification of all (complex) irreducible representations of a reductive group with connected centre, over a finite field. To achieve this, the author uses etale intersection cohomology, and detailed information on representations of Weyl groups.
At the crossroads of representation theory, algebraic geometry and finite group theory, this 2004 book blends together many of the main concerns of modern algebra, with full proofs of some …
At the crossroads of representation theory, algebraic geometry and finite group theory, this 2004 book blends together many of the main concerns of modern algebra, with full proofs of some of the most remarkable achievements in the area. Cabanes and Enguehard follow three main themes: first, applications of étale cohomology, leading to the proof of the recent Bonnafé–Rouquier theorems. The second is a straightforward and simplified account of the Dipper–James theorems relating irreducible characters and modular representations. The final theme is local representation theory. One of the main results here is the authors' version of Fong–Srinivasan theorems. Throughout the text is illustrated by many examples and background is provided by several introductory chapters on basic results and appendices on algebraic geometry and derived categories. The result is an essential introduction for graduate students and reference for all algebraists.
In some recent work, Lusztig outlined a generalisation of the construction of Deligne and Lusztig to reductive groups over finite rings coming from the ring of integers in a local …
In some recent work, Lusztig outlined a generalisation of the construction of Deligne and Lusztig to reductive groups over finite rings coming from the ring of integers in a local field, modulo some power of the maximal ideal. Lusztig conjectures that all irreducible representations of these groups are contained in the cohomology of a certain family of varieties. We show that, contrary to what was expected, there exist representations that cannot be realised by the varieties given by Lusztig. Moreover, we show how the remaining representations in the case under consideration can be realised in the cohomology of a different kind of variety. This may suggest a way to reformulate Lusztig's conjecture.
Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group …
Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such that $p$ is very good for $\mathbb{G}\times\mathbb{F}_{q}$, the groups $\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})$ and $\mathbb{G}(W_{2}(\mathbb{F}_{q}))$ have the same number of irreducible representations of dimension $d$, for each $d$. Equivalently, there exists an isomorphism of group algebras $\mathbb{C}[\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})]\cong\mathbb{C}[\mathbb{G}(W_{2}(\mathbb{F}_{q}))]$.
Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group …
Let $\mathbb{F}_{q}$ be a finite field of characteristic $p$, and let $W_{2}(\mathbb{F}_{q})$ be the ring of Witt vectors of length two over $\mathbb{F}_{q}$. We prove that for any reductive group scheme $\mathbb{G}$ over $\mathbb{Z}$ such that $p$ is very good for $\mathbb{G}\times\mathbb{F}_{q}$, the groups $\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})$ and $\mathbb{G}(W_{2}(\mathbb{F}_{q}))$ have the same number of irreducible representations of dimension $d$, for each $d$. Equivalently, there exists an isomorphism of group algebras $\mathbb{C}[\mathbb{G}(\mathbb{F}_{q}[t]/t^{2})]\cong\mathbb{C}[\mathbb{G}(W_{2}(\mathbb{F}_{q}))]$.
In this paper, we establish the Mackey formula for groupoids, extending the well-known formula in abstract groups context. This formula involves the notion of groupoid-biset, its orbit set and the …
In this paper, we establish the Mackey formula for groupoids, extending the well-known formula in abstract groups context. This formula involves the notion of groupoid-biset, its orbit set and the tensor product over groupoids, as well as cosets by subgroupoids.
We study the moduli space of principally polarized abelian varieties over fields of positive characteristic. In this paper we describe certain unions of Ekedahl-Oort strata contained in the supersingular locus …
We study the moduli space of principally polarized abelian varieties over fields of positive characteristic. In this paper we describe certain unions of Ekedahl-Oort strata contained in the supersingular locus in terms of Deligne-Lusztig varieties. As a corollary we show that each Ekedahl-Oort stratum contained in the supersingular locus is reducible except possibly for small <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
We give a criterion which determines when a union of one-dimensional Deligne-Lusztig varieties has a connected closure. We obtain a new, short proof of the connectedness criterion for Deligne-Lusztig varieties …
We give a criterion which determines when a union of one-dimensional Deligne-Lusztig varieties has a connected closure. We obtain a new, short proof of the connectedness criterion for Deligne-Lusztig varieties due to Lusztig.
After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using …
After some general remarks about characters of finite groups (possibly twisted by an automorphism), this chapter focuses on the generalised characters $R(T,\theta)$ which where introduced by Deligne and Lustzig using cohomological methods. We refer to the books by Carter and Digne-Michel for proofs of some fundamental properties, like orthogonality relations and degree formulae. Based on these results, we develop in some detail the basic formalism of Lusztig's book, which leads to a classification of the irreducible characters of finite groups of Lie type in terms of a fundamental Jordan decomposition. Using the general theory about regular embeddings in Chapter 1, we state and discuss that Jordan decomposition in complete generality, that is, without any assumption on the center of the underlying algebraic group. The final two sections give an introduction to the problems of computing Green functions and characteristic functions of character sheaves.
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After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig's result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of …
After collectiong some properties of irreducible representations of finite Coxeter groups we state and explain Lusztig's result on the decomposition of Deligne-Lusztig characters and then give a detailed exposition of the parametrisation and the properties of unipotent characters of finite reductive groups and related data like Fourier matrices and eigenvalues of Frobenius. We then describe the decomposition of Lusztig induction and collect the most recent results on its commutation with Jordan decomposition. We end the chapter with a survey of the character theory of finite disconnected reductive groups.
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Let $G$ be a finite group of Lie type. In order to determine the character table of $G$, Lusztig developed the theory of character sheaves. In this framework, one has …
Let $G$ be a finite group of Lie type. In order to determine the character table of $G$, Lusztig developed the theory of character sheaves. In this framework, one has to find the transformation between two bases for the space of class functions on $G$, one of them being the irreducible characters of $G$, the other one consisting of characteristic functions associated to character sheaves. In principle, this has been achieved by Lusztig and Shoji, but the underlying process involves some scalars which are still unknown in many cases. The problem of specifying these scalars can be reduced to considering cuspidal character sheaves. We will deal with the latter for the specific case where $G=E_7(q)$, and $q$ is a power of the bad prime $p=2$ for $E_7$.
This text consists of the introduction, table of contents, and bibliography of a long manuscript (703 pages) that is currently submitted for publication. This manuscript develops an extension of Garside's …
This text consists of the introduction, table of contents, and bibliography of a long manuscript (703 pages) that is currently submitted for publication. This manuscript develops an extension of Garside's approach to braid groups and provides a unified treatment for the various algebraic structures that appear in this context. The complete text can be found at http://www.math.unicaen.fr/~garside/Garside.pdf.
Inserted in the context of algebraic curves defined over finite fields, the present thesis addresses the study of the following three topics: plane sections of Fermat surfaces over finite fields; …
Inserted in the context of algebraic curves defined over finite fields, the present thesis addresses the study of the following three topics: plane sections of Fermat surfaces over finite fields; bounds for the number of F q -rational points on aX d Y d -X d -Y d + b = 0 and the number of chords of an affinely regular polygon inscribed in a hyperbola passing through a given point; the number of F q n -rational points, the L-polynomial and the automorphism group of the generalized Suzuki curve.
Abstract We give a description of the cohomology groups of the structure sheaf on smooth compactifications $$\overline{X}(w)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>w</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> …
Abstract We give a description of the cohomology groups of the structure sheaf on smooth compactifications $$\overline{X}(w)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>w</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> of Deligne–Lusztig varieties X ( w ) for $$\textrm{GL}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>GL</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> , for all elements w in the Weyl group. As a consequence, we obtain the $$\textrm{mod}\ p^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>mod</mml:mtext> <mml:mspace /> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> and integral p -adic étale cohomology of $$\overline{X}(w)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>w</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Moreover, using our result for $$\overline{X}(w)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>w</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> and a spectral sequence associated to a stratification of $$\overline{X}(w)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>X</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>w</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , we deduce the $$\textrm{mod}\ p^m$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>mod</mml:mtext> <mml:mspace /> <mml:msup> <mml:mi>p</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> </mml:math> and integral p -adic étale cohomology with compact support of X ( w ). In our proof of the main theorem, in addition to considering the Demazure–Hansen smooth compactifications of X ( w ), we show that a similar class of constructions provide smooth compactifications of X ( w ) in the case of $$\textrm{GL}_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>GL</mml:mtext> <mml:mi>n</mml:mi> </mml:msub> </mml:math> . Furthermore, we show in the appendix that the Zariski closure of X ( w ), for any connected reductive group G and any w , has pseudo-rational singularities.
Let G be an unramified reductive group over a nonarchimedian local field F .The so-called Langlands Fundamental Lemma is a family of conjectural identities between orbital integrals for G(F ) …
Let G be an unramified reductive group over a nonarchimedian local field F .The so-called Langlands Fundamental Lemma is a family of conjectural identities between orbital integrals for G(F ) and orbital integrals for endoscopic groups of G.In this paper we prove the Langlands fundamental lemma in the particular case where F is a finite extension of F p ((t)), G is a unitary group and p > rank(G).Waldspurger has shown that this particular case implies the Langlands fundamental lemma for unitary groups of rank < p when F is any finite extension of Q p .We follow in part a strategy initiated by Goresky, Kottwitz and MacPherson.Our main new tool is a deformation of orbital integrals which is constructed with the help of the Hitchin fibration for unitary groups over projective curves.Langlands et Shelstad (cf.[La-Sh]) ont défini un facteur de transfert ∆(γ, δ), qui est le produit d'un signe et de la puissance G ÉRARD LAUMON AND BAO CH ÂU NG ÔDans le chapitre 2, nous explicitons la construction de la fibration de Hitchin dans le cas du groupe unitaire.Nous faisons le lien entre la fibration Hitchin d'un groupe unitaire et la fibration de Hitchin d'un de ces groupes endoscopiques.Dans le chapitre 3, le coeur de ce travail, nous démontrons une identité globale, que l'on devrait pouvoir identifier à une identité globale qui apparaît dans la stabilisation de la formule des traces.L'énoncé principal de ce chapitre est le théorème 3.9.3.On le démontre à l'aide d'un isomorphisme en cohomologie équivariante.Comme nous l'avons mentionné plus haut, l'isomorphisme en cohomologie équivariante que nous construisons est analogue à celui construit antérieurement dans [G-K-M].Comme nous l'avons déjà dit notre construction s'appuie sur un énoncé de pureté, démontré dans le paragraphe 3.2, et d'un argument de déformation.Dans le chapitre 4, nous expliquons comment passer d'une situation locale donnée, à une situation globale du type de celle considérée dans le chapitre 3. Ici, l'outil de base est un théorème de Bertini rationnel 4.4.1, démontré par Gabber ([Gab]) et Poonen ([Poo]).Le comptage de la section (4.6) est analogue à celui du théorème (15.8) de [G-K-M].Enfin, dans un appendice, nous démontrons une variante A.1.2du théorème de localisation d'Atiyah-Borel-Segal. Puis nous présentons le calcul de la cohomologie équivariante d'un fibré en droites projective et d'un fibré en droites projectives pincées.Nous démontrons dans le dernier appendice une formule de points fixes.0.4.Précautions d 'emploi de nos résultats.Dans ce travail nous avons admis certains résultats sur la cohomologie -adique des champs algébriques.
Let G be a connected reductive algebraic group defined over a finite field k . The finite group G ( k ) of k -rational points of G acts on …
Let G be a connected reductive algebraic group defined over a finite field k . The finite group G ( k ) of k -rational points of G acts on the spherical building B ( G ), a polyhedron which is functorially associated with G . We identify the subspace of points of B ( G ) fixed by a regular semisimple element s of G ( k ) topologically as a subspace of a sphere (apartment) in B ( G ) which depends on an element of the Weyl group which is determined by s . Applications include the derivation of the values of certain characters of G ( k ) at s by means of Lefschetz theory. The characters considered arise from the action of G ( k ) on the cohomology of equivariant sheaves over B ( G ).
Abstract The preservation principle of the local theta correspondence predicts the existence of a chain of irreducible supercuspidal representations of p -adic classical groups. In this paper, we give an …
Abstract The preservation principle of the local theta correspondence predicts the existence of a chain of irreducible supercuspidal representations of p -adic classical groups. In this paper, we give an explicit characterization of the chain starting from an irreducible supercuspidal representations of a unitary group of one variable or an orthogonal group of two variables. In particular, we define the Lusztig-like correspondence of generic cuspidal data for p -adic groups and establish its relation with local theta correspondence of supercuspidal representations for p -adic dual pairs.
The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive …
The goal of this note is to show that in the case of ‘transversal intersections’ the ‘true local terms’ appearing in the Lefschetz trace formula are equal to the ‘naive local terms’. To prove the result, we extend the strategy used in our previous work, where the case of contracting correspondences is treated. Our new ingredients are the observation of Verdier that specialization of an étale sheaf to the normal cone is monodromic and the assertion that local terms are ‘constant in families’. As an application, we get a generalization of the Deligne–Lusztig trace formula.
James Arthur a récemment mis en lumière de nombreuses analogies entre des objets attachés à un groupe réductif p-adique et des objets attachés aux divers R-groupes (cf. [Ar2, en particulier …
James Arthur a récemment mis en lumière de nombreuses analogies entre des objets attachés à un groupe réductif p-adique et des objets attachés aux divers R-groupes (cf. [Ar2, en particulier Remarks (2) p. 118]). Nous nous intéressons ici au cas d'un groupe réductif fini, i.e., du groupe G F des points fixes sous un endomorphisme de Frobenius F d'un groupe G algébrique réductif connexe sur une clôture algébrique d'un corps fini F q et défini sur 픽 q . Le rôle des R-groupes est joué ici par les groupes de ramification $$ W_{\text{G}^F } (\text{M,}\,\sigma \text{)}\,\text{ = }\,\text{\{ }r\, \in \,\text{N}_{\text{G}^F } (\text{M)/M}^F |{}^r\sigma = \sigma \} $$ associés à des paires (M, σ) formées d'un sous-groupe de Levi F-stable M d'un sous-groupe parabolique F-stable de G et d'une représentation irréductible cuspidale σ de M F ; ces groupes sont des extensions centrales de groupes de Coxeter.
On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to …
On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis–Alvis duality map and Mackey's theorem and the results that can be deduced from it, before moving on to a discussion of Deligne–Lusztig induction and Lusztig's Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples.
Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich …
Through the fundamental work of Deligne and Lusztig in the 1970s, further developed mainly by Lusztig, the character theory of reductive groups over finite fields has grown into a rich and vast area of mathematics. It incorporates tools and methods from algebraic geometry, topology, combinatorics and computer algebra, and has since evolved substantially. With this book, the authors meet the need for a contemporary treatment, complementing in core areas the well-established books of Carter and Digne–Michel. Focusing on applications in finite group theory, the authors gather previously scattered results and allow the reader to get to grips with the large body of literature available on the subject, covering topics such as regular embeddings, the Jordan decomposition of characters, d-Harish–Chandra theory and Lusztig induction for unipotent characters. Requiring only a modest background in algebraic geometry, this useful reference is suitable for beginning graduate students as well as researchers.
Let G be a split connected reductive algebraic group over Q_p such that both G and its dual group G-hat have connected centres. Motivated by a hypothetical p-adic Langlands correspondence …
Let G be a split connected reductive algebraic group over Q_p such that both G and its dual group G-hat have connected centres. Motivated by a hypothetical p-adic Langlands correspondence for G(Q_p) we associate to an n-dimensional ordinary (i.e. Borel valued) representation rho : Gal(Q_p-bar/Q_p) to G-hat(E) a unitary Banach space representation Pi(rho)^ord of G(Q_p) over E that is built out of principal series representations. (Here, E is a finite extension of Q_p.) Our construction is inspired by the "ordinary part" of the tensor product of all fundamental algebraic representations of G. There is an analogous construction over a finite extension of F_p. In the latter case, when G=GL_n we show under suitable hypotheses that Pi(rho)^ord occurs in the rho-part of the cohomology of a compact unitary group. We also prove a weaker version of this result in the p-adic case.
<!-- *** Custom HTML *** --> Hall-Littlewood functions and Green functions associated to complex reflection groups $W = G(r, 1, n)$ were constructed in [S1] by means of symbols, which …
<!-- *** Custom HTML *** --> Hall-Littlewood functions and Green functions associated to complex reflection groups $W = G(r, 1, n)$ were constructed in [S1] by means of symbols, which are a generalization of partitions. In this paper, we consider such functions in the case where the symbols are of very special type, called "limit symbols". The situation becomes simple, and is close to the case of symmetric groups when the symbols tends to the "limit". In the case where $W$ is a Weyl group of type $B_n$, we give a closed formula for Hall-Littlewood functions, and verify some of the conjectures stated in [S1] for the case of Green functions attached to limit symbols.
Let GL and U denote the finite general linear and unitary groups extended by the transpose inverse automorphism, respectively, where q is a power of the prime p. Let n …
Let GL and U denote the finite general linear and unitary groups extended by the transpose inverse automorphism, respectively, where q is a power of the prime p. Let n be odd, and let χ be an irreducible character of either of these groups which is an extension of a real-valued character of GL or U. Let yτ be an element of GL or U such that (yτ)2 is regular unipotent in GL or U, respectively. We show that is prime to p and otherwise. Several intermediate results on real conjugacy classes and real-valued characters of these groups are obtained along the way.
(1977). Green polynomials op finite classical groups. Communications in Algebra: Vol. 5, No. 12, pp. 1241-1258.
(1977). Green polynomials op finite classical groups. Communications in Algebra: Vol. 5, No. 12, pp. 1241-1258.
As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro--Tiep and Sp\"ath for simple groups …
As a step to establish the blockwise Alperin weight conjecture for all finite groups, we verify the inductive blockwise Alperin weight condition introduced by Navarro--Tiep and Sp\"ath for simple groups of Lie type $\mathsf A$, split or twisted. Key to the proofs is to reduce the verification of the inductive condition to the isolated (that means unipotent) blocks, using the Jordan decomposition for blocks of finite reductive groups given by Bonnaf\'e, Dat and Rouquier.
The Alperin weight conjecture is central to the modern representation theory of finite groups, and it is still open, despite many different approaches from different points of view. This paper …
The Alperin weight conjecture is central to the modern representation theory of finite groups, and it is still open, despite many different approaches from different points of view. This paper surveys methods and results relating the Alperin weight conjecture, especially the recent developments in the inductive investigation.