DOCSLIB.ORG

Lie Groups and Algebraic Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Lie Groups and Algebraic Groups Lie groups and algebraic groups M S RAGHUNATHAN and T N VENKATARAMANA∗ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India. ∗e-mail: [email protected] We give an exposition of certain topics in Lie groups and algebraic groups. This is not a complete survey and is meant to highlight certain aspects with which the authors are familiar, and to which there has been a significant contribution from mathematicians working in India. 1. Introduction linear classical groups GLn(R), GLn(C), SOn(R), SOn(C), Spn(R)andSpn(C). Let us call a con- One of the fundamental branches of mathematics nected Lie group simple if G has no connected is the study of symmetries of any mathemati- normal closed subgroups other than G and the cal object (like a geometric object, or a differ- identity. It is a remarkable fact that simple ential/polynomial equation, etc.). The symmetries Lie groups can be completely classified; they form a group, and group theoretical properties of are the special linear groups, orthogonal groups this group give us information on other mathemati- and symplectic groups. Apart from these, the cal properties of the object under consideration. list is a finite one (the so-called exceptional For example, the solvability of the ‘Galois group’ groups). This is the Cartan–Killing classification, of a polynomial equation is equivalent to the solva- whichnowadays,isdescribedintermsofthe bility of the equation by radicals. A similar state- ‘Dynkin diagrams’. ment can be made for solvability of differential A related class of groups are simple algebraic equations. groups. They can be defined over an arbitrary field. The most ubiquitous class of groups is that of Lie For a discussion on classification of these groups, groups. These are manifolds (geometrical objects, see section 5. functions on which can be differentiated) such that the group operations are infinitely differentiable. Lie groups occur naturally in geometry, analysis, 3. Representation theory algebra and physics. Prototypical examples are the groups GLn(R)(GLn(C)) of invertible n×n matri- 3.1 Finite dimensional representations ces with real (complex) entries. By its very nature, a Lie group has analytic, geo- Lie groups are essentially (up to coverings) closed metric and algebraic properties, each of which is subgroups of GLn(R). One of the fundamental a highly developed field of study. These fields are problems in the theory of Lie groups was to find all interrelated and each of these fields contributes to possible ways in which a Lie group can be (embed- the other. ded as) a subgroup of GLn(R)(orGLn(C)). This is a problem which is completely solved for connected 2. Examples and classification Lie groups. The foregoing problem is equivalent to finding We first give some examples of Lie groups. all possible group homomorphisms G → GLn(C)– The most frequently occurring ones are the with varying n – which are infinitely differentiable Keywords. Lie groups; representations; algebraic groups; arithmetic groups, rigidity. 627 628 M S RAGHUNATHAN AND T N VENKATARAMANA (smooth). Such homomorphisms are called (finite representations. The explicit realisation is due to dimensional) ‘representations’ of G.Inother Borel and Weil. words, elements of G may be viewed as invertible An important problem in physics is the decom- linear transformations of the vector space Cn.By position of a representation ρ of a compact Lie an abuse of notation, the vector space Cn together group restricted to a compact subgroup H.In with the G-action, is also called a representation particular, the decomposition of a tensor product of G. is especially important; the PRV conjecture (see We will describe – briefly – the solution of this [12]) describes exactly what happens in this case problem, for compact connected Lie groups. A com- for certain important constituents of the tensor pact Lie group G may be shown (with some work, product; the conjecture was proved by Shrawan using the theory of compact operators) to be none Kumar [25]. In a different direction, a unique fac- other than a closed subgroup of the unitary group torisation theorem for tensor products was proved t U(n)ofn × n-matrices g ∈ GLn(C)with gg =1, by C S Rajan [22]. for some n.IfG is connected, it is possible to replace G by a conjugate in U(n) such that the 3.2 Infinite dimensional representations intersection with G of the group of diagonals in U(n) is a maximal connected abelian subgroup T We will now consider unitary representations of G of G (it is called a ‘maximal torus’ in G). All maxi- i.e. infinite dimensional Hilbert spaces H on which mal tori in G are conjugate (this purely algebraic non-compact Lie groups G act preserving the inner statement was first proved by using the Lefschetz product. A special class of Lie groups are those fixed point theorem, which is a fundamental whose Lie algebra is a direct sum of irreducible result in algebraic topology; this is an instance (finite dimensional) representations of G each of of geometric properties of G yielding algebraic which has dimension strictly greater than one. Such information). groups are called semi-simple Lie groups. It is no Direct sums and tensor products of representa- longer true that a representation is a direct ‘sum’ tions may be defined in an obvious way. A repre- of irreducibles. The direct sum has to be replaced sentation is said to be irreducible, if it cannot be by a direct integral. written as a direct sum of two representations. If A basic example of a representation is the space 2 G is compact, every representation can be written L (G) of the space of square integrable functions as a direct sum of irreducible ones. Therefore, the on G (square integrable with respect to the Haar problem of describing representations reduces to measure). Even in this case, a description of the that of irreducible representations. It can be shown direct integral decomposition is a formidably diffi- that if ρ : G → GLn(C) is irreducible, the trace cult problem. It was however, completely solved by function g → trace(ρ(g)) on G completely deter- the great Indian mathematician Harish-Chandra in mines the representation. aseriesofpapers. To continue, one may show that the quotient The description of representations which occur, manifold G/T is a compact complex manifold such together with the number of times they occur (the that the group G (which acts on G/T by trans- multiplicity, or the Plancherel measure in the case lations), acts by holomorphic automorphisms. If χ of direct integrals) in the direct integral decomposi- 2 is a character of T (a continuous homomorphism tion of L (G) is called the Plancherel theorem. The of T into U(1) = the circle group), then one may classification of these representations contributing form a ‘line bundle’ Lχ on G/T which can be to the Plancherel theorem is in terms of fundamen- shown to be a holomorphic line bundle. The space tal building blocks, which are called ‘discrete series 0 H (G/T, Lχ) of holomorphic sections can be shown representations’. Harish-Chandra proved that a to be finite dimensional, on which G acts by left semi-simple Lie group G has discrete series if translations. One has the fundamental theorem of and only if G has a connected maximal abelian Hermann Weyl, Armand Borel and Andre Weil: subgroup T which is compact (a compact Car- tan subgroup T ). Harish-Chandra gave a formula Theorem 1. The representation ρ = ρχ = (the Plancherel measure) for the representations, 0 2 H (G/T, Lχ) is irreducible. Moreover, every irre- together with multiplicities, which occur in L (G). ducible representation of G may be realised in this The Plancherel measure for G/K (K amaxi- way, for some character χ of T . The trace function mal compact subgroup of G) was computed for trace(ρ(g)) can be completely described in terms many classical groups by T S Bhanu-Murthy [4]. of χ. The complete classification of the representations occurring in L2(G) was achieved by Knapp and Historically, Hermann Weyl first described the Zuckerman [5]. irreducible representations in terms of their Somewhat similar to the case of compact Lie traces, without an explicit construction of the groups, Harish Chandra gave a formula – without LIE GROUPS AND ALGEBRAIC GROUPS 629 explicitly constructing the discrete series rep- of projective space which preserves its ‘geometry’ resentations – for the ‘trace distribution’ of (i.e. incidence relations such as subspaces of a given discrete series. The problem of explicitly con- vector space) is actually a linear isomorphism. The structing the discrete series was solved by various proof is a repeated use of the high transitivity people. Many Indian mathematicians contributed of the action by the group of linear isomorphisms to this development: an algebraic construction was on the ‘geometry’ of the projective space. Similarly given by K R Parthasarathy, Ranga Rao and theorems of affine geometry translate into pro- V S Varadarajan. A description, closer in spirit to perties of the affine transformation group. Similar the Borel–Weil (–Bott) theorem, was conjectured results hold for other geometries. by Langlands, and proved by M S Narasimhan– Felix Klein had the idea that extension of these Okamoto (in the case when the symmetric space results to other geometries can be deduced from of the group G is a Hermitian symmetric domain) group theoretical properties of certain transfor- and by R Parthasarathy (for G arbitrary but the mation groups associated to these geometries. He weight of the discrete series is sufficiently regu- sketched a programme (the Erlangen programme) lar).
Load more

Recommended publications
  • Arxiv:2003.06292V1 [Math.GR] 12 Mar 2020 Eggnrtr N Ignlmti.Tedaoa Arxi Matrix Diagonal the Matrix
    ALGORITHMS IN LINEAR ALGEBRAIC GROUPS SUSHIL BHUNIA, AYAN MAHALANOBIS, PRALHAD SHINDE, AND ANUPAM SINGH ABSTRACT. This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decompositionwith respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups. 1. INTRODUCTION Spinor norm was first defined by Dieudonné and Kneser using Clifford algebras. Wall [21] defined the spinor norm using bilinear forms. These days, to compute the spinor norm, one uses the definition of Wall. In this paper, we develop a new definition of the spinor norm for split and twisted orthogonal groups. Our definition of the spinornorm is rich in the sense, that itis algorithmic in nature. Now one can compute spinor norm using a Gaussian elimination algorithm that we develop in this paper. This paper can be seen as an extension of our earlier work in the book chapter [3], where we described Gaussian elimination algorithms for orthogonal and symplectic groups in the context of public key cryptography. In computational group theory, one always looks for algorithms to solve the word problem. For a group G defined by a set of generators hXi = G, the problem is to write g ∈ G as a word in X: we say that this is the word problem for G (for details, see [18, Section 1.4]). Brooksbank [4] and Costi [10] developed algorithms similar to ours for classical groups over finite fields.
    [Show full text]
  • Dynamics for Discrete Subgroups of Sl 2(C)
    DYNAMICS FOR DISCRETE SUBGROUPS OF SL2(C) HEE OH Dedicated to Gregory Margulis with affection and admiration Abstract. Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups [30]: \A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title Theory of discrete subgroups of SL2(C)". In this article, we will discuss a few recent advances regarding this missing topic from his book, which were influenced by his earlier works. Contents 1. Introduction 1 2. Kleinian groups 2 3. Mixing and classification of N-orbit closures 10 4. Almost all results on orbit closures 13 5. Unipotent blowup and renormalizations 18 6. Interior frames and boundary frames 25 7. Rigid acylindrical groups and circular slices of Λ 27 8. Geometrically finite acylindrical hyperbolic 3-manifolds 32 9. Unipotent flows in higher dimensional hyperbolic manifolds 35 References 44 1. Introduction A discrete subgroup of PSL2(C) is called a Kleinian group. In this article, we discuss dynamics of unipotent flows on the homogeneous space Γn PSL2(C) for a Kleinian group Γ which is not necessarily a lattice of PSL2(C). Unlike the lattice case, the geometry and topology of the associated hyperbolic 3-manifold M = ΓnH3 influence both topological and measure theoretic rigidity properties of unipotent flows. Around 1984-6, Margulis settled the Oppenheim conjecture by proving that every bounded SO(2; 1)-orbit in the space SL3(Z)n SL3(R) is compact ([28], [27]).
    [Show full text]
  • Arxiv:Math/0511624V1
    Automorphism groups of polycyclic-by-finite groups and arithmetic groups Oliver Baues ∗ Fritz Grunewald † Mathematisches Institut II Mathematisches Institut Universit¨at Karlsruhe Heinrich-Heine-Universit¨at D-76128 Karlsruhe D-40225 D¨usseldorf May 18, 2005 Abstract We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homotopy self-equivalences of K(Γ, 1)-spaces and obtain an extension of arithmeticity results of Sullivan in rational homo- topy theory. 2000 Mathematics Subject Classification: Primary 20F28, 20G30; Secondary 11F06, 14L27, 20F16, 20F34, 22E40, 55P10 arXiv:math/0511624v1 [math.GR] 25 Nov 2005 ∗e-mail: [email protected] †e-mail: [email protected] 1 Contents 1 Introduction 4 1.1 Themainresults ........................... 4 1.2 Outlineoftheproofsandmoreresults . 6 1.3 Cohomologicalrepresentations . 8 1.4 Applications to the groups of homotopy self-equivalences of spaces 9 2 Prerequisites on linear algebraic groups and arithmetic groups 12 2.1 Thegeneraltheory .......................... 12 2.2 Algebraicgroupsofautomorphisms. 15 3 The group of automorphisms of a solvable-by-finite linear algebraic group 17 3.1 The algebraic structure of Auta(H)................
    [Show full text]
  • Stabilizers of Lattices in Lie Groups
    Journal of Lie Theory Volume 4 (1994) 1{16 C 1994 Heldermann Verlag Stabilizers of Lattices in Lie Groups Richard D. Mosak and Martin Moskowitz Communicated by K. H. Hofmann Abstract. Let G be a connected Lie group with Lie algebra g, containing a lattice Γ. We shall write Aut(G) for the group of all smooth automorphisms of G. If A is a closed subgroup of Aut(G) we denote by StabA(Γ) the stabilizer of Γ n n in A; for example, if G is R , Γ is Z , and A is SL(n;R), then StabA(Γ)=SL(n;Z). The latter is, of course, a lattice in SL(n;R); in this paper we shall investigate, more generally, when StabA(Γ) is a lattice (or a uniform lattice) in A. Introduction Let G be a connected Lie group with Lie algebra g, containing a lattice Γ; so Γ is a discrete subgroup and G=Γ has finite G-invariant measure. We shall write Aut(G) for the group of all smooth automorphisms of G, and M(G) = α Aut(G): mod(α) = 1 for the group of measure-preserving automorphisms off G2 (here mod(α) is thegcommon ratio measure(α(F ))/measure(F ), for any measurable set F G of positive, finite measure). If A is a closed subgroup ⊂ of Aut(G) we denote by StabA(Γ) the stabilizer of Γ in A, in other words α A: α(Γ) = Γ . The main question of this paper is: f 2 g When is StabA(Γ) a lattice (or a uniform lattice) in A? We point out that the thrust of this question is whether the stabilizer StabA(Γ) is cocompact or of cofinite volume, since in any event, in all cases of interest, the stabilizer is discrete (see Prop.
    [Show full text]
  • Matrix Lie Groups
    Maths Seminar 2007 MATRIX LIE GROUPS Claudiu C Remsing Dept of Mathematics (Pure and Applied) Rhodes University Grahamstown 6140 26 September 2007 RhodesUniv CCR 0 Maths Seminar 2007 TALK OUTLINE 1. What is a matrix Lie group ? 2. Matrices revisited. 3. Examples of matrix Lie groups. 4. Matrix Lie algebras. 5. A glimpse at elementary Lie theory. 6. Life beyond elementary Lie theory. RhodesUniv CCR 1 Maths Seminar 2007 1. What is a matrix Lie group ? Matrix Lie groups are groups of invertible • matrices that have desirable geometric features. So matrix Lie groups are simultaneously algebraic and geometric objects. Matrix Lie groups naturally arise in • – geometry (classical, algebraic, differential) – complex analyis – differential equations – Fourier analysis – algebra (group theory, ring theory) – number theory – combinatorics. RhodesUniv CCR 2 Maths Seminar 2007 Matrix Lie groups are encountered in many • applications in – physics (geometric mechanics, quantum con- trol) – engineering (motion control, robotics) – computational chemistry (molecular mo- tion) – computer science (computer animation, computer vision, quantum computation). “It turns out that matrix [Lie] groups • pop up in virtually any investigation of objects with symmetries, such as molecules in chemistry, particles in physics, and projective spaces in geometry”. (K. Tapp, 2005) RhodesUniv CCR 3 Maths Seminar 2007 EXAMPLE 1 : The Euclidean group E (2). • E (2) = F : R2 R2 F is an isometry . → | n o The vector space R2 is equipped with the standard Euclidean structure (the “dot product”) x y = x y + x y (x, y R2), • 1 1 2 2 ∈ hence with the Euclidean distance d (x, y) = (y x) (y x) (x, y R2).
    [Show full text]
  • Debashish Goswami Jyotishman Bhowmick Quantum Isometry Groups Infosys Science Foundation Series
    Infosys Science Foundation Series in Mathematical Sciences Debashish Goswami Jyotishman Bhowmick Quantum Isometry Groups Infosys Science Foundation Series Infosys Science Foundation Series in Mathematical Sciences Series editors Gopal Prasad, University of Michigan, USA Irene Fonseca, Mellon College of Science, USA Editorial Board Chandrasekhar Khare, University of California, USA Mahan Mj, Tata Institute of Fundamental Research, Mumbai, India Manindra Agrawal, Indian Institute of Technology Kanpur, India S.R.S. Varadhan, Courant Institute of Mathematical Sciences, USA Weinan E, Princeton University, USA The Infosys Science Foundation Series in Mathematical Sciences is a sub-series of The Infosys Science Foundation Series. This sub-series focuses on high quality content in the domain of mathematical sciences and various disciplines of mathematics, statistics, bio-mathematics, financial mathematics, applied mathematics, operations research, applies statistics and computer science. All content published in the sub-series are written, edited, or vetted by the laureates or jury members of the Infosys Prize. With the Series, Springer and the Infosys Science Foundation hope to provide readers with monographs, handbooks, professional books and textbooks of the highest academic quality on current topics in relevant disciplines. Literature in this sub-series will appeal to a wide audience of researchers, students, educators, and professionals across mathematics, applied mathematics, statistics and computer science disciplines. More information
    [Show full text]
  • Some Finiteness Results for Groups of Automorphisms of Manifolds 3
    SOME FINITENESS RESULTS FOR GROUPS OF AUTOMORPHISMS OF MANIFOLDS ALEXANDER KUPERS Abstract. We prove that in dimension =6 4, 5, 7 the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of Rn and various types of automorphisms of 2-connected manifolds. Contents 1. Introduction 1 2. Homologically and homotopically finite type spaces 4 3. Self-embeddings 11 4. The Weiss fiber sequence 19 5. Proofs of main results 29 References 38 1. Introduction Inspired by work of Weiss on Pontryagin classes of topological manifolds [Wei15], we use several recent advances in the study of high-dimensional manifolds to prove a structural result about diffeomorphism groups. We prove the classifying spaces of such groups are often “small” in one of the following two algebro-topological senses: Definition 1.1. Let X be a path-connected space. arXiv:1612.09475v3 [math.AT] 1 Sep 2019 · X is said to be of homologically finite type if for all Z[π1(X)]-modules M that are finitely generated as abelian groups, H∗(X; M) is finitely generated in each degree. · X is said to be of finite type if π1(X) is finite and πi(X) is finitely generated for i ≥ 2. Being of finite type implies being of homologically finite type, see Lemma 2.15. Date: September 4, 2019. Alexander Kupers was partially supported by a William R.
    [Show full text]
  • Some Groups of Finite Homological Type
    Some groups of finite homological type Ian J. Leary∗ M¨ugeSaadeto˘glu† June 10, 2005 Abstract For each n ≥ 0 we construct a torsion-free group that satisfies K. S. Brown’s FHT condition and is FPn, but is not FPn+1. 1 Introduction While working on comparing different notions of Euler characteristic, K. S. Brown introduced a new homological finiteness condition for discrete groups [5, IX.6]. The group G is said to be of finite homological type or FHT if G has finite virtual cohomological dimension, and for every G-module M whose underlying abelian group is finitely generated, the homology groups Hi(G; M) are all finitely generated. If G is FHT , then one may define a ‘na¨ıve Euler characteristic’ for every finite-index subgroup H of G, as the alternating sum of the dimensions of the homology groups of H with rational coefficients. One question that arises is the connection between FHT and the usual homological finiteness conditions FP and FPn, which were introduced by J.-P. Serre [8]. (We shall define these conditions below.) It is easy to see that any group G of type FP is FHT , and one might conjecture that every torsion-free group that is FHT is also of type FP . The aim of this paper is to show that this is not the case. For each n ≥ 0, we exhibit a torsion-free group Gn that is FHT and of type FPn, but that is not of type FPn+1. Our construction is based on R. Bieri’s construction of a group that is FPn but not FPn+1 [2, Prop 2.14].
    [Show full text]
  • A Theorem on Discrete Groups and Some Consequences of Kazdan's
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF FUNCTIONAL ANALYSIS 6, 203-207 (1970) A Theorem on Discrete Groups and Some Consequences of Kazdan’s Thesis ROBERT R. KALLMAN* Department of Mathematics, Yale University, New Haven, Connecticut 06520 Communicated by Irving Segal Received June 21, 1969 Let G be a noncompact simple Lie group, and let r be a discrete subgroup of G such that G/P has finite volume. The main result of this paper is that the left ring of P is a Type II1 von Neumann algebra. This result in turn is applied to solve, in some generality, a problem in dynamical systems posed by I. M. Gelfand. INTRODUCTION If G is a locally compact unimodular group, let Y(G) be the von Neumann algebra generated by left translations L(a) (a in G) on L2(G). Let I’ be a discrete subgroup of G, and let p be a measure on G/r which is invariant under the natural action of G. The measure class of p is uniquely determined. If f is in L2(G/F), let (V4f)W = flu-’ - 4. The following problem is of interest in dynamical systems (cf. Gelfand [3]). Let &?(G, r) be the von Neumann algebra on L2(G) @ L2(G/.F) generated by L(a) @ V(u) (u in G) and I @ M(4) (here 4 is in L”(G/r, p) and (&Z(+)f)(x) = ~$(x)f(x), for all f in L2(G/r)).
    [Show full text]
  • On Dense Subgroups of Homeo +(I)
    On dense subgroups of Homeo+(I) Azer Akhmedov Abstract: We prove that a dense subgroup of Homeo+(I) is not elementary amenable and (if it is finitely generated) has infinite girth. We also show that the topological group Homeo+(I) does not satisfy the Stability of the Generators Property, moreover, any finitely subgroup of Homeo+(I) admits a faithful discrete representation in it. 1. Introduction In this paper, we study basic questions about dense subgroups of Homeo+(I) - the group of orientation preserving homeomorphisms of the closed interval I = [0; 1] - with its natural C0 metric. The paper can be viewed as a continuation of [A1] and [A2] both of which are devoted to the study of discrete subgroups of Diff+(I). Dense subgroups of connected Lie groups have been studied exten- sively in the past several decades; we refer the reader to [BG1], [BG2], [Co], [W1], [W2] for some of the most recent developments. A dense subgroup of a Lie group may capture the algebraic and geometric con- tent of the ambient group quite strongly. This capturing may not be as direct as in the case of lattices (discrete subgroups of finite covolume), but it can still lead to deep results. It is often interesting if a given Lie group contains a finitely generated dense subgroup with a certain property. For example, finding dense subgroups with property (T ) in the Lie group SO(n + 1; R); n ≥ 4 by G.Margulis [M] and D.Sullivan [S], combined with the earlier result of Rosenblatt [Ro], led to the brilliant solution of the Banach-Ruziewicz Problem for Sn; n ≥ 4.
    [Show full text]
  • On the Third-Degree Continuous Cohomology of Simple Lie Groups 2
    ON THE THIRD-DEGREE CONTINUOUS COHOMOLOGY OF SIMPLE LIE GROUPS CARLOS DE LA CRUZ MENGUAL ABSTRACT. We show that the class of connected, simple Lie groups that have non-vanishing third-degree continuous cohomology with trivial ℝ-coefficients consists precisely of all simple §ℝ complex Lie groups and of SL2( ). 1. INTRODUCTION ∙ Continuous cohomology Hc is an invariant of topological groups, defined in an analogous fashion to the ordinary group cohomology, but with the additional assumption that cochains— with values in a topological group-module as coefficient space—are continuous. A basic refer- ence in the subject is the book by Borel–Wallach [1], while a concise survey by Stasheff [12] provides an overview of the theory, its relations to other cohomology theories for topological groups, and some interpretations of algebraic nature for low-degree cohomology classes. In the case of connected, (semi)simple Lie groups and trivial real coefficients, continuous cohomology is also known to successfully detect geometric information. For example, the situ- ation for degree two is well understood: Let G be a connected, non-compact, simple Lie group 2 ℝ with finite center. Then Hc(G; ) ≠ 0 if and only G is of Hermitian type, i.e. if its associated symmetric space of non-compact type admits a G-invariant complex structure. If that is the case, 2 ℝ Hc(G; ) is one-dimensional, and explicit continuous 2-cocycles were produced by Guichardet– Wigner in [6] as an obstruction to extending to G a homomorphism K → S1, where K<G is a maximal compact subgroup. The goal of this note is to clarify a similar geometric interpretation of the third-degree con- tinuous cohomology of simple Lie groups.
    [Show full text]
  • A NEW APPROACH to UNRAMIFIED DESCENT in BRUHAT-TITS THEORY by Gopal Prasad
    A NEW APPROACH TO UNRAMIFIED DESCENT IN BRUHAT-TITS THEORY By Gopal Prasad Dedicated to my wife Indu Prasad with gratitude Abstract. We present a new approach to unramified descent (\descente ´etale") in Bruhat-Tits theory of reductive groups over a discretely valued field k with Henselian valuation ring which appears to be conceptually simpler, and more geometric, than the original approach of Bruhat and Tits. We are able to derive the main results of the theory over k from the theory over the maximal unramified extension K of k. Even in the most interesting case for number theory and representation theory, where k is a locally compact nonarchimedean field, the geometric approach described in this paper appears to be considerably simpler than the original approach. Introduction. Let k be a field given with a nontrivial R-valued nonarchimedean valuation !. We will assume throughout this paper that the valuation ring × o := fx 2 k j !(x) > 0g [ f0g of this valuation is Henselian. Let K be the maximal unramified extension of k (the term “unramified extension" includes the condition that the residue field extension is separable, so the residue field of K is the separable closure κs of the residue field κ of k). While discussing Bruhat-Tits theory in this introduction, and beginning with 1.6 everywhere in the paper, we will assume that ! is a discrete valuation. Bruhat- Tits theory of connected reductive k-groups G that are quasi-split over K (i.e., GK contains a Borel subgroup, or, equivalently, the centralizer of a maximal K-split torus of GK is a torus) has two parts.
    [Show full text]