DOCSLIB.ORG

Cocycle Growth for the Steinberg Representation§

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Cocycle Growth for the Steinberg Representation§ Cocycle growth for the Steinberg representation THÈSE NO 6599 (2016) PRÉSENTÉE LE 26 FÉVRIER 2016 À LA FACULTÉ DES SCIENCES DE BASE CHAIRE DE THÉORIE ERGODIQUE ET GÉOMÉTRIQUE DES GROUPES PROGRAMME DOCTORAL EN MATHÉMATIQUES ÉCOLE POLYTECHNIQUE FÉDÉRALE DE LAUSANNE POUR L'OBTENTION DU GRADE DE DOCTEUR ÈS SCIENCES PAR Thibaut DUMONT acceptée sur proposition du jury: Prof. K. Hess Bellwald, présidente du jury Prof. N. Monod, directeur de thèse Prof. B. Klingler, rapporteur Dr Y. de Cornulier, rapporteur Prof. D. Testerman, rapporteuse Suisse 2016 Amafamille.` Thibaut Dumont Ecole´ polytechnique f´ed´eralede Lausanne, 1015 Lausanne, Switzerland. E-mail : [email protected] Url : math.dumont.ch 2010 Mathematics Subject Classification.—20J06, 22E41, 20E42. The author was supported by the Advanced Grant 267635 Rigidity of the European Research Council attributed to Prof. Nicolas Monod, and by the Swiss Confederation. COCYCLE GROWTH FOR THE STEINBERG REPRESENTATION Thibaut Dumont Abstract.— This thesis investigates the growth of the natural cocycle introduced by Klingler for the Steinberg representation. When possible, we extend the framework of simple algebraic groups over a local field to arbitrary Euclidean buildings. In rank one, the growth of the cocycle is determined to be sublinear. In higher rank, the complexity of the problem leads us to study the geometry of A2 buildings, where we describe in detail the relative position of three points. Key words and phrases.—Group theory, cohomology, continuous cohomology, building, Stein- berg representation. R´esum´e.— Cette th`ese ´etudie la croissance du cocycle naturel pour le module de Steinberg. Nous ´etendons les travaux de Klingler dans le cas des groups alg´ebriques simples sur un corps local aux immeubles euclidiens lorsque cela est possible. En rang un, la croissance du cocycle de Klingler est sous-lin´eaire.En rang sup´erieur la complexit´edelaquestionnousentrainedansl’´etude de la g´eom´etrie des immeubles A2,o`u nous d´ecrivons en d´etail la position relative de trois points. Mots clefs.—Th´eorie des groupes, cohomologie, cohomologie continue, immeuble, representa- tion de Steinberg. ACKNOWLEDGEMENT I would like to express my deep gratitude to Prof. Nicolas Monod for taking me as his PhD student and for accompanying me during this journey. Working with him has been a real pleasure full of amazement. My admiration is endless. We thank the jury members, the president, Prof. Kathryn Hess Bellwald, the internal examiner, Prof. Donna Testerman, and the two external examiners, Dr. Yves de Cornulier and Prof. Bruno Klingler, for accepting to be part of it with the duties it implies. My thanks go to the various people whom I had the opportunity to discuss the present work with. In particular, I warmly thank Prof. Pierre-Emmanuel Caprace who helped me complete some of the arguments of Section 5.2, Kristian K. Olesen for his help with some LATEX issues, and Guillaume Blanc for taking five minutes to provide the Python script for the proof of Proposition 5.3.13. To everyone who supported me during my time at EPFL, colleagues, friends, family, I thank youfromthebottomofmyheart. Lausanne, February 2, 2016. CONTENTS Acknowledgement.......................................................................... vii Introduction................................................................................ 1 Klingler’s cocycle and the Steinberg representation. ...................................... 1 1. Preliminaries............................................................................ 5 1.1. Geometryoflocallycompactgroups................................................... 5 1.2. Buildingsandgroups.................................................................. 8 1.3. Groupcohomology.................................................................... 17 2. The Steinberg representation and the natural cocycle of Klingler................ 21 2.1. Irreducible admissible representations................................................. 21 2.2. Natural cocycle for the Steinberg representation....................................... 23 2.3. Unitarizability via the Poisson transform.............................................. 27 3. Building theoretic formulation........................................................ 33 3.1. TheboundaryΩ....................................................................... 34 3.2. Topologicalindependenceofthebasepoint............................................ 42 3.3. Factorisation of volX throughlargespheres............................................ 44 3.4. VisualmeasuresandthePoissontransform............................................ 48 4. The rank one case...................................................................... 55 4.1. HomogeneoustreesandextendedPoissontransform................................... 55 4.2. Strategyandresults................................................................... 59 4.3. The case A............................................................................ 62 4.4. The case B............................................................................ 74 5. Geometry of A2-buildings.............................................................. 83 5.1. Retractionsandarrowings............................................................. 83 5.2. Convexhullofthreepoints............................................................ 87 5.3. Relativedirections..................................................................... 93 Notation index.............................................................................109 Subject index...............................................................................111 Bibliography................................................................................113 INTRODUCTION The study of cocycle growth for isometric linear representations is a fine cohomological tool. Bounded cohomology is a good example of a fundamental cohomology theory with a growth con- dition, see [Mon06] for instance. The growth of 1-cocycles for unitary representations of locally compact groups relates to renowned properties such as Kazhdan’s Property (T) or Haagerup Prop- erty, see [BHV08]and[CTV07]. The present work investigates the growth of a particular cocycle in the following setting. Let G be a compactly generated locally compact group with an associated word length distance dS, where S is a compact generating set, and let V be an isometric linear representation of G in a complex Banach space (V,·). Studying the growth of a G-equivariant n-cocycle c : Gn+1 → V amounts to look for possible bounds of c(g0,...,gn) n+1 as (g0,...,gn) vary in a subset of G , and, preferably, depending on the distances between the variables g0,...,gn ∈ G. In particular, bounded cohomology treats with cocycles bounded uni- formly on the whole Gn+1.In[CTV07], the authors look at 1-cocycles of unitary representations that are unbounded, but having some upper bound depending on dS(g0,g1), allowing the bound to only take place outside a compact subset of G2. The main motivation for the present research is a problem posed by Monod. In [MS04], he and Shalom performed a procedure, called ‘quasification’, of an unbounded 1-cocycle, yielding a new bounded cohomology class,called the median class for groups acting on a tree. The median class has a natural generalization for various groups acting on CAT(0) cube complexes, see [CFI12]. In [Mon06, Problem P] Monod asks if a similar quasification could be applied to the cocycles defined by Klingler in [Kli03]. The first natural step toward an answer is to determine their growth, which is the central objectif of this thesis. Klingler’s cocycle and the Steinberg representation Let G be the group of F -points of a connected, simply connected, almost F -simple algebraic group over a local field F of characteristic zero. It is a compactly generated totally disconnected locally compact group, which possesses an interesting pre-unitary admissible representation called n+1 the Steinberg representation St. Klingler built in [Kli03]anaturalcocyclevolG : G → St for 2 INTRODUCTION this G-module generating the continuous cohomology in degree n equal the F -rank of G, i.e. n · Hc (G, St)=C [volG]. The aim of our research is to determine the growth of volG. In rank n = 1, we obtain the following result, (Theorem 4.2.4 and Corollary 4.2.5). Theorem.—Let G be SL2(F ) and dS be a word metric on G. There exists a constant C>0, depending only on the cardinality of the residue field(1) of F , such that volG(g0,g1)St ≤ C · dS(g0,g1), for all g0,g1 ∈ G. In [GJ15], Gournay and Jolissaint independently obtained a finer and more general estimate, which proves our bound to be asymptotically sharp. Nevertheless, we hope that our proof shed light on the combinatorics of Klingler’s cocycle. In higher rank, the difficulty is significantly increased. The definition of the Klingler’s cocycle volG, as well as the Steinberg representation of G,are closely related to the Bruhat-Tits building of G. The latter is a locally finite irreducible Euclidean building X on which G acts by type preserving automorphism. It can be endowed with a proper CAT(0) metric d for which G acts by isometries. In the preliminary Chapter 1, we recall that the geometries of (X, d)and(G, dS) are roughly identical, (Proposition 1.1.8), and moreover that the cohomology of G can be fully understood from that of X by considering G-equivariant cocycles on X. In turn, the Bruhat-Tits building X is sufficient to study the cohomology of G and the growth
Load more

Recommended publications
  • Simple Supercuspidals and the Langlands Correspondence
    Simple supercuspidals and the Langlands correspondence Benedict H Gross April 23, 2020 Contents 1 Formal degrees2 2 The conjecture2 3 Depth zero for SL2 3 4 Simple supercuspidals for SL2 4 5 Simple supercuspidals in the split case5 6 Simple wild parameters6 7 The trace formula7 8 Function fields9 9 An irregular connection 10 10 `-adic representations 12 11 F -isocrystals 13 12 Conclusion 13 1 1 Formal degrees In the fall of 2007, Mark Reeder and I were thinking about the formal degrees of discrete series repre- sentations of p-adic groups G. The formal degree is a generalization of the dimension of an irreducible representation, when G is compact. It depends on the choice of a Haar measure dg on G: if the irre- ducible representation π is induced from a finite dimensional representation W of an open, compact R subgroup K, then the formal degree of π is given by deg(π) = dim(W )= K dg. Mark had determined the formal degree in several interesting cases [19], including the series of depth zero supercuspidal representations that he had constructed with Stephen DeBacker [2]. We came across a paper by Kaoru Hiraga, Atsushi Ichino, and Tamotsu Ikeda, which gave a beautiful conjectural for- mula for the formal degree in terms of the L-function and -factor of the adjoint representation of the Langlands parameter [13, x1]. This worked perfectly in the depth zero case [13, x3.5], and we wanted to test it on more ramified parameters, where the adjoint L-function is trivial. Naturally, we started with the group SL2(Qp).
    [Show full text]
  • Sheaves on Buildings and Modular Representations of Chevalley Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA %, 319-346 (1985) Sheaves on Buildings and Modular Representations of Chevalley Groups MARK A. RONAN AND STEPHEND. SMITH* Department of Mathematics, University of Illinois at Chicago, Chicago, Illinois 60680 Communicated by Walter Feit Received October 23, 1983 INTRODUCTION Let A be the building (see Tits [24]) for a Chevalley group G of rank n; it is a simplicial complex of dimension n - 1. The homology of A was studied by Solomon and Tits [lS], who showed in particular that the top homology group H, _ ,(A) affords the Steinberg representation for G. In [9], Lusztig constructed discrete-series representations using homology with a special system of coefficients, in the natural characteristic, on buildings of type A,. Later Deligne and Lusztig [7; see also 1 and 51 obtained a “duality” between certain representations, using coefficient homology. These coefficient systems may be described as fixed-point sheaves, FV (see Section I), where V is a G-module. In this paper we are primarily concerned with modules and sheaves over the natural field k of definition of the group G. (In [ 1, 5, 71, the modules are in characteristic zero.) The motivation for our work comes from several areas: (1) Simple groups. Emerging techniques of Timmesfeld and others (see [23, 15, 221) often lead to problems requiring the “local” recognition of kG-modules-from information given only at the parabolic subgroups (which define the building A). We shall see in (1.3) that such modules are quotients of the zero-homology group HO of a suitable coefficient system.
    [Show full text]
  • Steinberg Representations and Harmonic Cochains for Split Adjoint
    Steinberg representations and harmonic cochains for split adjoint quasi-simple groups Y. A¨ıt Amrane 3 juin 2021 Abstract Let G be an adjoint quasi-simple group defined and split over a non-archimedean local field K. We prove that the dual of the Steinberg representation of G is isomor- phic to a certain space of harmonic cochains on the Bruhat-Tits building of G. The Steinberg representation is considered with coefficients in any commutative ring. Introduction Let K be a non-archimedean local field. Let G be the K-rational points of a reductive K-group of semi-simple rank l. Let T be a maximal K-split torus in G and let P be a minimal parabolic K-subgroup of G that contains T . There is an abuse of language because we mean the K-rational points of these algebraic subgroups of G. For a commutative ring arXiv:1708.00717v1 [math.RT] 2 Aug 2017 M, the Steinberg representation of G with coefficients in M is the M[G]-module : ∞ C (G/P, M) St(M)= ∞ Q C (G/Q, M) where Q runs through all the parabolic subgroupsP of G containing P . l−1 In [4], A. Borel and J.-P. Serre, computed the reduced cohomology group H˜ (Yt, M) of the topologized building Yt of the parabolic subgroups of G and proved that we have an isomorphism of M[G]-modules : l−1 ∼ H˜ (Yt, M) = St(M). 1 Then they ”added” this building at infinity to the Bruhat-Tits building X of G to get X compactified to a contractible space Zt = X ∐ Yt.
    [Show full text]
  • Arxiv:1404.0861V1 [Math.RT] 3 Apr 2014 St Td the Study to Is Rbe Foei Neetdi Opeecaatrinformat Character Complete in Interested Understood
    NOTES ON REPRESENTATIONS OF FINITE GROUPS OF LIE TYPE DIPENDRA PRASAD These are the notes of some lectures given by the author for a workshop held at TIFR in December, 2011, giving an exposition of the Deligne-Lusztig theory. The aim of these lectures was to give an overview of the subject with several examples without burdening them with detailed proofs of the theorems involved for which we refer to the original paper of Deligne and Lusztig, as well as the beautiful book of Digne and Michel on the subject. The author thanks Shripad M. Garge for writing and texing the first draft of these notes. Contents 1. Preliminaries on Algebraic groups 2 2. Classical groups and tori 3 3. Some Exceptional groups 5 4. Basic notions in representation theory 6 5. The Deligne-Lusztig theory 7 6. Unipotent cuspidal representations 9 7. The Steinberg representation 9 8. Dimension of the Deligne-Lusztig representation 10 9. The Jordan decomposition of representations 11 10. Exercises 12 11. References 13 arXiv:1404.0861v1 [math.RT] 3 Apr 2014 The Deligne-Lusztig theory constructs certain (virtual) representations of G(Fq) denoted as RG(T, θ) where G is a connected reductive algebraic group defined over Fq, T is a maximal torus in G defined ∼ × over F and θ is a character of T (F ), θ : T (F ) C× Q . The representation R (T, θ) is called q q q → −→ l G the Deligne-Lusztig induction of the character θ of T (Fq) to G(Fq), generalizing parabolic induction with which it coincides when T is a maximally split maximal torus in G.
    [Show full text]
  • Representation Theory and Number Theory
    REPRESENTATION THEORY AND NUMBER THEORY Contents 1. Tuesday September 20 2 1.1. The local Langlands conjectures 2 1.2. Some examples 3 1.3. Unramified representations/parameters 3 1.4. Other examples 4 1.5. Wildly ramified representations of SU3(E=k) 5 2. Tuesday September 27 6 2.1. Review of algebraic number theory: global theory 6 2.2. Galois theory 7 2.3. Artin's theory 8 2.4. Review of algebraic number theory: local theory 9 3. October 4, 2011 10 3.1. Why did Artin introduce his L-functions? 10 3.2. Some calculations 12 3.3. What about function fields? 13 4. Tuesday October 11 14 4.1. Special values of Artin L-functions 14 4.2. How not to prove the Riemann hypothesis 16 4.3. Back to special values 16 4.4. Galois representations not over C 18 5. Tuesday October 18 18 5.1. Reductive groups 19 5.2. Tori 19 5.3. Roots 19 5.4. Coroots 20 5.5. (reduced) Root datum 20 6. Tuesday October 25 21 6.1. Classification of reductive groups over an arbitrary field 24 7. Tuesday November 1 25 7.1. The classification of forms 25 7.2. The L-group 26 8. November 18, 2011 28 8.1. Local parameters 28 8.2. Unramified representations of p-adic groups 29 9. November 15, 2011 30 9.1. Reformulation of a Langlands parameter 30 9.2. Unramified representations of p-adic groups 31 9.3. The Satake isomorphism 32 9.4. Global representations 32 Date: Fall 2011.
    [Show full text]
  • Character Tables of the General Linear Group and Some of Its Subroups
    Character Tables of the General Linear Group and Some of its Subroups By Ayoub Basheer Mohammed Basheer [email protected] [email protected] Supervisor : Professor Jamshid Moori [email protected] School of Mathematical Sciences University of KwaZulu-Natal, Pietermaritzburg, South Africa A project submitted in the fulfillment of the requirements for the Masters Degree of Science in the School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg. November 2008 Abstract The aim of this dissertation is to describe the conjugacy classes and some of the ordinary irreducible characters of the finite general linear group GL(n; q); together with character tables of some of its subgroups. We study the structure of GL(n; q) and some of its important subgroups such as SL(n; q);UT (n; q); SUT (n; q);Z(GL(n; q));Z(SL(n; q)); GL(n; q)0 ; SL(n; q)0 ; the Weyl group W and parabolic subgroups Pλ: In addition, we also discuss the groups P GL(n; q); P SL(n; q) and the affine group Aff(n; q); which are related to GL(n; q): The character tables of GL(2; q); SL(2; q); SUT (2; q) and UT (2; q) are constructed in this dissertation and examples in each case for q = 3 and q = 4 are supplied. A complete description for the conjugacy classes of GL(n; q) is given, where the theories of irre- ducible polynomials and partitions of i 2 f1; 2; ··· ; ng form the atoms from where each conjugacy class of GL(n; q) is constructed.
    [Show full text]
  • Categorical Actions from Lusztig Induction and Re- Striction on finite General Linear Groups
    Categorical actions from Lusztig induction and re- striction on finite general linear groups O. Dudas, M. Varagnolo, E. Vasserot∗ Abstract. In this note we explain how Lusztig’s induction and restriction functors yield categorical actions of Kac-Moody algebras on the derived category of unipotent representations. We focus on the example of finite linear groups and induction/restriction associated with split Levi subgroups, providing a derived analogue of Harish-Chandra induction/restriction as studied by Chuang-Rouquier in [5]. 2010 Mathematics Subject Classification. Primary 20C33. Keywords. Finite reductive groups; Deligne-Lusztig theory; Higher representation the- ory. Introduction Let Gn = GLn(q) be the general linear group over a finite field with q elements. In order to construct and study the representations of Gn with coefficients in a field k of positive characteristic ℓ ∤ q, it is common to consider the chain of subgroups {1} = G0 ⊂ G1 ⊂ ··· ⊂ Gn ⊂ · · · together with the Harish-Chandra induction and restriction functors − also called parabolic induction and restriction − F - kGn-mod m kGn+1-mod. E But there is more to it: Chuang-Rouquier showed in [5] that for any m ≥ 1, the functor Fm is endowed with a natural action of an affine Hecke algebra of type Am−1 controlling most of the behaviour of these functors. From this action they defined a family of i-induction and i-restriction functors {Fi} and {Ei} inducing, at the level of Grothendieck groups, an action of a Kac-Moody algebra g. This led to the notion of categorical g-action on the category of (unipotent) representations for various Gn (see [18]).
    [Show full text]
  • Jacquet Modules of Parabolically Induced Representations and Weyl Groups
    JACQUET MODULES OF PARABOLICALLY INDUCED REPRESENTATIONS AND WEYL GROUPS DUBRAVKA BAN Abstract. The representation parabolically induced from an ir- reducible supercuspidal representation is considered. Irreducible components of Jacquet modules with respect to induction in stages are given. The results are used for consideration of generalized Steinberg representations. 1. Introduction Jacquet modules of parabolically induced representations can be applied to some problems in representation theory, for example, the question of reducibility of parabolically induced representations ([T3], [J3]). They are also used for important work on description of dis- crete series for classical p-adic groups ([MT], [J1], [J2]). These articles concern the classical groups Sp(n, F ) and SO(2n + 1,F ). For calcu- lating Jacquet modules of parabolically induced representations, they are using a structure on representations of the groups Sp(n, F ) and SO(2n + 1,F ), described in [T1]. The purpose of this article is to find new techniques for calculating Jacquet modules for any connected p-adic group. Consequently, we also describe the structure of parabolically induced representations, their irreducible subrepresentations and irreducible subquotients. Let G be a connected reductive p-adic group, P = MU a standard parabolic subgroup of G and σ an irreducible supercuspidal represen- tation of M. The geometric lemma ([BZ], [C], here Theorem 2.1) de- scribes composition factors of rM,G ◦ iG,M (σ), where iG,M denotes functor of parabolic induction and rM,G the Jacquet functor ([BZ]). After eliminating all zero components in rM,G ◦ iG,M (σ) ([BZ], [C], here Theorem 2.2), we prove some interesting facts about the structure of iG,M (σ) and rM,G ◦ iG,M (σ) (Lemma 3.1 and Corollary 4.3).
    [Show full text]
  • Jacquet Modules and Induced Representations∗
    Jacquet modules and induced representations∗ Marko Tadi´c 1. Introduction In this paper, we shall review some possible applications of Jacquet mod- ules to the study of parabolically induced representations of reductive p-adic groups. Let us start with a few brief remarks about the history of Jacquet mod- ules. Jacquet modules do not make their appearance in representation theory until the end of the 1960’s. Using them, H. Jacquet was able to obtain some important basic results in the representation theory of p-adic groups (e.g., his subrepresentation theorem and the equivalence of the different charac- terizations of cuspidal [Jc]). These results are still somewhat qualitative in nature. To use Jacquet modules as a calculational tool, more quantitative results are needed. A major step in this direction was taken by W. Cassel- man, who calculated the Jordan-H¨older series for the Jacquet module of an induced representation, among other things ([Cs]). A significant number of his results were obtained independently by J. Bernstein and A.V. Zelevinsky ([B-Z]). Before going on to discuss how one can use Jacquet modules to study reducibility questions for induced representations, let us indicate some limi- tations to this approach. For example, consider SL(2). If χ2 = 1 with χ =6 1, SL(2) the representation IndP∅ (χ) is reducible. However, one cannot obtain this reducibility through a simple use of Jacquet modules. A similar situation SL(2) occurs with IndP∅ (1), except that in this case the representation is irre- ∗This paper has been published in Mathematical Communications vol.
    [Show full text]
  • Smooth Representations of P-Adic Reductive Groups
    SMOOTH REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS FLORIAN HERZIG These are notes for my expository talk at the Instructional Conference on Representation Theory and Arithmetic at Northwestern University, May 5–9, 2008. 1. Motivation Smooth representations of p-adic groups arise in number theory mainly through the study of automorphic representations, and thus in the end, for example, from modular forms. We saw in the first lecture by Matt Emerton that a modular form, thought of as function on the set of lattices with level N structure, we obtain a ∞ function in C (GL2(Z)\GL2(R) × GL2(Z/N), C) satisfying certain differ- ential equations (coming from holomorphicity and the weight of the modular form) and growth conditions (related to the cusps). As was suggested, this space has an adelic interpretation: by the strong approximation theorem, GL2(AQ) = GL2(Q)GL2(Zb)GL2(R), and a simple argument shows that ∞ ∼ ∞ C (GL2(Z)\GL2(R) × GL2(Z/N), C) = C (GL2(Q)\GL2(AQ)/U(N), C), where U(N) = ker(GL2(Zb) GL2(Z/N)). Taking the union of these spaces over all N ≥ 1 (still with holomorphicity and growth conditions imposed), we ∞ ∞ get a big “automorphic” space A ⊂ C (GL2(Q)\GL2(AQ), C) which has ∞ a natural action of GL2( ) by right translations. This action is “smooth”: AQ by definition, every element of A∞ is fixed by some open subgroup, namely U(N) for some N. Moreover if we start with a cuspidal newform f, we can get a subrep- resentation π∞ ⊂ A∞ by restricting ourselves to all oldforms of f in each level.
    [Show full text]
  • Is a Method for Constructing Representations Ρ : G → GL(V ) Where V Is a Vector Space and G Is a finite Lie Group
    DELIGNE–LUSZTIG THEORY ANDREJ VUKOVIĆ Abstract. We show how Deligne–Lusztig theory is used to produce irreducible represen- tations of SL2(Fq). We begin by discussing sheaves and schemes, eventually moving on to the theory of group schemes and reductive linear algebraic groups. We look at the represen- tation theory of SL2(Fq) and show how it runs into difficulties that are resolved by a clever technique of Drinfeld. We then explain how Deligne and Lusztig generalized this technique to other finite groups of Lie type and show that their method really does coincide with Drinfeld’s in the particular case of SL2(Fq). 1. Sheaves and Schemes Deligne–Lusztig theory, first developed in [3], is a method for constructing representations ρ : G ! GL(V ) where V is a vector space and G is a finite Lie group. These ideas, which the namesake authors have written about from 1976 onwards, heavily relies on results about `-adic cohomology, a tool developed by Alexander Grothendieck and many others throughout the 1960s. To discuss Deligne–Lusztig theory therefore requires an introduction to algebraic geometry. We will have to assume knowledge of category theory (e.g., functors, pullbacks, equalizers), commutative algebra (e.g., radical and homogeneous ideals), and representation theory (e.g., induced representations), and Lie theory (the content of this course) but will define the new terminology from algebraic geometry as it comes up. Because of the amount of algebraic geometry required to begin defining the basic objects of Deligne–Lusztig theory, we will omit proofs with impunity. The material of the current section can be found in any textbook on modern algebraic ge- ometry.
    [Show full text]
  • Homological Vanishing for the Steinberg Representation
    Homological vanishing for the Steinberg representation Avner Ash Andrew Putman∗ Steven V Sam† November 20, 2017 Abstract For a field k, we prove that the ith homology of the groups GLn(k), SLn(k), Sp2n(k), SOn,n(k), and SOn,n+1(k) with coefficients in their Steinberg representations vanish for n ≥ 2i + 2. 1 Introduction Let G be a connected reductive group over a field k. A basic geometric object associated to G(k) is its Tits building. By definition, this is the simplicial complex TG(k) whose i-simplices are increasing sequences 0 ( P0 ( ··· ( Pi ( G(k) of parabolic k-subgroups of G(k). Letting r be the semisimple k-rank of G, the complex TG(k) is (r − 1)-dimensional, and the Solomon–Tits theorem [Br1, Theorem IV.5.2] says that in fact TG(k) is homotopy equivalent to a wedge of (r − 1)-dimensional spheres. Letting R be a commutative ring, the Steinberg representation of G(k) over R, denoted StG(k; R), is He r−1(TG(k); R). This is one of the most important representations of G(k); for instance, if G is any of the classical groups in Theorem 1.1 below (e.g. G = SLn) and k is a finite field of characteristic p, then StG(k; C) is the unique irreducible representation of G(k) whose dimension is a positive power of p (see [MalZ], which proves this aside from three small cases that must be checked by hand). See [H] for a survey of many results concerning the Steinberg representation.
    [Show full text]