DOCSLIB.ORG

On Representations Distinguished by Unitary Groups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Representations Distinguished by Unitary Groups ON REPRESENTATIONS DISTINGUISHED BY UNITARY GROUPS BROOKE FEIGON, EREZ LAPID AND OMER OFFEN To the memory of Jonathan Rogawski Abstract. Let E=F be a quadratic extension of number fields. We study periods and reg- ularized periods of cusp forms and Eisenstein series on GLn(AE ) over a unitary group of a Hermitian form with respect to E=F . We provide factorization for these periods into locally defined functionals, express these factors in terms of suitably defined local periods and charac- terize global distinction. We also study in detail the analogous local question and analyze the space of invariant linear forms under a unitary group. Contents Index 2 Introduction 3 Part 1. Local Theory 7 1. Notation and preliminaries 7 2. Bessel distributions 15 3. Local identities of Bessel distributions 20 4. Open local periods 25 5. Closed periods 31 6. Applications of the geometric lemma 37 7. Local Bessel identities { the split case 43 8. Local Bessel identities { the unramified case 47 9. Dependence on additive character 50 Appendix A. Inner product for induced representations of GLn, by Erez Lapid and Omer Offen 54 Appendix B. Distinguished representations in the Archimedean case, by Avraham Aizenbud and Erez Lapid 60 Part 2. Global Theory 63 10. Factorization of unitary periods { cuspidal representations 63 11. Factorization of unitary periods { Eisenstein series 67 Appendix C. The relative trace formula, by Erez Lapid and Omer Offen 74 Part 3. Local Applications 76 12. Local Bessel identities and functional equations { the general case 76 13. Multiplicity of unitary periods 82 Appendix D. Upper semicontinuity of multiplicity, by Erez Lapid and Omer Offen 88 References 89 Date: May 4, 2012. The second and third named authors were partially supported by grants from the Israel Science Foundation. 1 Index ∗ w :τ (aM ) , 32 C , 73 ∗ ~ 0 0 (aM )+, 9 w (π ), 22 AG, 64 w~(π), 21 CM (w : σ; λ; ), 13 Fw, 38 0 D ! D , 21 !V , 9 E(g; '; λ), 68 rl(σ; λ), 69 G◦[x], 27 f^(π), 14 H, 8 Rpi(G), 13 I(D; λ), 15 I, ≺, 38 G IM (T ), 11 θ, 31 G G λ 0 , 21 IM (σ), IM (σ; λ), 11 J(' : x; σ; λ), 70 cs(σ; λ), 13, 68 JM (' : x; α, λ), 26 δQ, 8 0 0 L(s; π1 × π2), "(s; π1 × π2; ), γ(s; π1 × π2; ), 13 B(f : σ ; λ), 19 L(x), R(y), 9 Bπ0 , 18, 65 M(w; σ; λ), M(w; W(σ); λ), 12 θ~, 12 0 N(w; σ; λ), N(w; W(σ); λ), 13 n(σ ; λ), Jσ0 (x; α, λ), 29 P , M , U , 7 kgk, 16 0 0 0 − − − Pκ, Mκ, Uκ, 7 P , M, U , 8 Qx, 7 κ−, 8 − X[x], G0[x], 20 σ−, λ , 12 [·; ·]π, 54 BIM;x, BIM , 24 A, 3 Px, 3 | AM,σ, 68 π · χ, π[s], π , 10 M ∗ _ ∆0 , 8 π , π , 10 ∗ EG(X; π ), 17 F(' : x; β; λ), 36 Z(' : x; β; λ), 32 0, , 7 π 0 δg , 11 0, 0, 11 LQ(σ; λ), 14 R(G), 10 x Φ ! f 0, f ! f 0, 20 ρ, 9 Φ α, 17 σ[Z], 14 ΦM , 28 σ1 × · · · × σt, 11 0 0 0 0 0 0 supp (π), 14 Πcusp,Πusqr,Πsqr,Πtemp,Πgen,Π , 11 c τ;ti τ;an S(Y ), 9 Π ,Π , 22 t 0 0 t 0 0−1 τ ∗ 0 ( U0×U0; 0× 0 ) Πcusp,Πusqr,Πsqr,Πtemp,Πgen, Π, Π , 11 S (G ) , 19 −1 0 ∗ (U0; ) Πbc-gen, 22 S (X) 0 , 21 τ π0 Πreg, 84 α , BIx, BI, 23 0 Πunr, 14 B~(σ ; λ), 31 W , 8 1Γ, 1Q, 9 W M (L), 8 µ ' ξ,Γ (ν; µ), 11 W [M], M(w), 38 L;M 2 ' , 11 Ξ, 69 M ('; σ; λ), 12, 68 a∗ , a , a∗, a , 8 W Q Q 0 0 π, Wπ, 64 ai, 21 W W (π), 11 α ◦ T , 17 W(π0), 65 A(σ; λ), 68 W(π), 18 A , 64 π w(x), 7 B(π), b ,[B](π), [b] , 21 π π W(' : σ; λ), 12, 68 bc, 3, 21 f ∗ Φ, 9 ; , 15 E C rL;M , rM , 11 `;`^ BD (f), 15 wκ, 8 ~ DL(%; λ), 37 wσ, 12 % M β , 36 w0 , 8 X◦, 25 x • g, 7 G0, G, X, 3 G0±, 19 π0 XL, 31 rl0 , 65 !π, 10 η, 3 D· χ, 15 D◦, 15 C, 64 2 ON REPRESENTATIONS DISTINGUISHED BY UNITARY GROUPS 3 Introduction Let E=F be a quadratic extension of number fields with Galois involution τ and let A = AF be the ring of adeles of F: Denote by η : F ∗nA∗ ! {±1g the quadratic character associated to E=F 0 by class field theory. Let G = GLn =F and let G be the restriction of scalars from E to F of GLn =E. Let X = fg 2 G : tgτ = gg be the F -subvariety of Hermitian matrices in G: Here tg is the transpose of a matrix g: For x 2 X(F ) denote by Gx = fg 2 G : tgτ xg = xg the unitary group (defined over F ) determined by x: The unitary period integral Px(φ) of a cusp form φ of G(A)(= GLn(AE)) is defined by Z Px(φ) = φ(h) dh: x x G (F )nG (A) A cuspidal automorphic representation π of G(A) is called Gx-distinguished if there exists a cusp form φ in the space of π such that Px(φ) 6= 0: The study of unitary period integrals has seen much progress in recent years. In particular, Jacquet showed that an irreducible cuspidal automorphic representation π of G(A) is in the image of quadratic base change bc if and only if π is Gx- distinguished for some x 2 X(F ) [Jac05a]. In fact, it suffices to take the quasi-split unitary group [Jac10]. In order to obtain these results, Jacquet, in the course of many years, invented and developed the relative trace formula on G(A) and together with his collaborators compared it to a Kuznetsov trace formula on G0(A) (see [Ye88, Ye89, JY90, Jac92, JY92, Ye93, Ye94, Jac95, Ye95, JY96, Jac98, Ye98, JY99, Jac01, Jac02, Jac03a, Jac03b, Jac04b, Jac05a, Jac05b, Lap06, Jac10]). This comparison of trace formulas gives additional information about the period integrals. To describe it, we first recall that by Arthur-Clozel, an irreducible cuspidal automorphic representa- tion π of G(A) is in the image of quadratic base change if and only if π is Galois invariant ([AC89]). In this case, π is the image of exactly two irreducible cuspidal representations of G0(A) of the form π0, π0 ⊗ η. We say that E=F splits at infinity if every Archimedean place of F splits in E: When n = 3; x = e and E=F splits at infinity, Jacquet showed in [Jac01, Theorem 1] that the period integral Px is factorizable on π and that its local components are governed by certain identities 0 0 between Bessel distributions associated to the local components πv of π and πv of π. The restric- tion to the case n = 3 was made since at the time the necessary relative trace identity was only available in that case. Based on recent developments (mainly [Jac03b, Jac04b, Jac05a, Lap06]), the method and result of Jacquet can be generalized to any n and x (Theorem 10.2). We also remove the assumption that E=F splits at infinity thanks to a recent result by Aizenbud and Gourevitch ([AG]) extending smooth transfer to the Archimedean case. As pointed out in [Jac01], the factorization of the global unitary period is far from formal since there is no multiplicity one for the local invariant functionals. For instance, for a quadratic extension of p-adic fields, any irreducible unramified principal series representation admits 2n−1 linearly independent invariant functionals with respect to a given unitary group. The failure of multiplicity one reflects the fact that locally, quadratic base change is not one-to-one, even up to a twist by η. Along with the factorization we provide for any x 2 X(F ) a criterion for an irreducible, Galois-invariant, cuspidal automorphic representation π = bc(π0) of G(A) to be Gx-distinguished. Namely, π is Gx-distinguished except for purely local obstructions depending on the (finitely many) 0 x 0 local components of π (rather than π) at the places where G is not quasi-split. (We recall that πv is generally not determined by πv, even up to a quadratic twist.) In the non-Archimedean case (for 0 0 which non-quasi-split unitary groups exist only for n even) the obstruction is that πv ' πv ⊗ ηv. The Archimedean obstruction is a little more complicated but it is still given explicitly in terms 0 0 0 0 0 0 0 of πv. More precisely, let w~ (πv) be the largest k such that we can write πv = δ × σ where δ is 0 0 a representation of GL2k(Fv) such that δ ' δ ⊗ η and × denotes parabolic induction. Then we have the following characterization. 4 BROOKE FEIGON, EREZ LAPID AND OMER OFFEN Theorem 0.1. Let x be a Hermitian form of rank n with respect to E=F and let π be an irreducible x cuspidal representation of GLn(AE). Then π is G -distinguished if and only if there exists a 0 0 cuspidal representation π of GLn(AF ) such that π = bc(π ) (i.e., π is Galois invariant) and 0 0 w~ (πv) is not bigger than the Witt index of x with respect to Ev=Fv for all inert places v of F over which Gx is not quasi-split.
Load more

Recommended publications
  • 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]
  • GEOMETRY and GROUPS These Notes Are to Remind You of The
    GEOMETRY AND GROUPS These notes are to remind you of the results from earlier courses that we will need at some point in this course. The exercises are entirely optional, although they will all be useful later in the course. Asterisks indicate that they are harder. 0.1 Metric Spaces (Metric and Topological Spaces) A metric on a set X is a map d : X × X → [0, ∞) that satisfies: (a) d(x, y) > 0 with equality if and only if x = y; (b) Symmetry: d(x, y) = d(y, x) for all x, y ∈ X; (c) Triangle Rule: d(x, y) + d(y, z) > d(x, z) for all x, y, z ∈ X. A set X with a metric d is called a metric space. For example, the Euclidean metric on RN is given by d(x, y) = ||x − y|| where v u N ! u X 2 ||a|| = t |an| n=1 is the norm of a vector a. This metric makes RN into a metric space and any subset of it is also a metric space. A sequence in X is a map N → X; n 7→ xn. We often denote this sequence by (xn). This sequence converges to a limit ` ∈ X when d(xn, `) → 0 as n → ∞ . A subsequence of the sequence (xn) is given by taking only some of the terms in the sequence. So, a subsequence of the sequence (xn) is given by n 7→ xk(n) where k : N → N is a strictly increasing function. A metric space X is (sequentially) compact if every sequence from X has a subsequence that converges to a point of X.
    [Show full text]
  • Forms of the Affine Line and Its Additive Group
    Pacific Journal of Mathematics FORMS OF THE AFFINE LINE AND ITS ADDITIVE GROUP PETER RUSSELL Vol. 32, No. 2 February 1970 PACIFIC JOURNAL OF MATHEMATICS Vol. 32, No. 2, 1970 FORMS OF THE AFFINE LINE AND ITS ADDITIVE GROUP PETER RUSSELL Let k be a field, Xo an object (e.g., scheme, group scheme) defined over k. An object X of the same type and isomorphic to Xo over some field K z> k is called a form of Xo. If k is 1 not perfect, both the affine line A and its additive group Gtt have nontrivial sets of forms, and these are investigated here. Equivalently, one is interested in ^-algebras R such that K ®k R = K[t] (the polynomial ring in one variable) for some field K => ky where, in the case of forms of Gα, R has a group (or co-algebra) structure s\R—>R®kR such that (K®s)(t) = £ ® 1 + 1 ® ί. A complete classification of forms of Gα and their principal homogeneous spaces is given and the behaviour of the set of forms under base field extension is studied. 1 If k is perfect, all forms of A and Gα are trivial, as is well known (cf. 1.1). So assume k is not perfect of characteristic p > 0. Then a nontrivial example (cf. [5], p. 46) of a form of Gα is the subgroup of Gα = Spec k[x, y] defined by yp = x + axp where aek, agkp. We show that this example is quite typical (cf. 2.1): Every form of Gtt pn is isomorphic to a subgroup of G« defined by an equation y = aQx + p pm atx + + amx , cii ek, aQΦ 0.
    [Show full text]
  • A STUDY on the ALGEBRAIC STRUCTURE of SL 2(Zpz)
    A STUDY ON THE ALGEBRAIC STRUCTURE OF SL2 Z pZ ( ~ ) A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Evan North April 2015 Contents 1 Introduction 1 2 Background 5 2.1 Group Theory . 5 2.2 Linear Algebra . 14 2.3 Matrix Group SL2 R Over a Ring . 22 ( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 Order of the Matrix Groups . 26 3.2 Conjugacy Classes of GL2 Fp ....................... 28 3.2.1 Linear Case . .( . .) . 29 3.2.2 First Quadratic Case . 29 3.2.3 Second Quadratic Case . 30 3.2.4 Third Quadratic Case . 31 3.2.5 Classes in SL2 Fp ......................... 33 3.3 Splitting of Classes of(SL)2 Fp ....................... 35 3.4 Results of SL2 Fp ..............................( ) 40 ( ) 2 4 Toward Lifting to SL2 Z p Z 41 4.1 Reduction mod p ...............................( ~ ) 42 4.2 Exploring the Kernel . 43 i 4.3 Generalizing to SL2 Z p Z ........................ 46 ( ~ ) 5 Closing Remarks 48 5.1 Future Work . 48 5.2 Conclusion . 48 1 Introduction Symmetries are one of the most widely-known examples of pure mathematics. Symmetry is when an object can be rotated, flipped, or otherwise transformed in such a way that its appearance remains the same. Basic geometric figures should create familiar examples, take for instance the triangle. Figure 1: The symmetries of a triangle: 3 reflections, 2 rotations. The red lines represent the reflection symmetries, where the trianlge is flipped over, while the arrows represent the rotational symmetry of the triangle.
    [Show full text]
  • On the Commutator Subgroups of Certain Unitary Groups
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OFALGEBRA 27, 306-310(1973) On the Commutator Subgroups of Certain Unitary Groups c. T. C. WALL Department of Mathematics, University of Liverpool, Liverpool, England Communicated by J. Tits Received February 29, 1972 Let G be a classical group, i.e., [8] for some field K, central simple algebra A of finite dimension over K, and (anti-) involution a: of A, G = {X E A: x% = l}. Then it is wellknown (for references see [3, Chapter II]) that, with a handful of small exceptions, if a: has index v 3 1, and T is the subgroup of G generated by transvections then T modulo its center (= G n K) is simple. It is less easy to identify the subgroup T; the results depend on 01.If a: is of symplectic type, then T = G [3, p. 481. If 01is of unitary type, and A is trivial (i.e., a matrix ring over K), then (with one exception) T is the subgroup of G of matrices with determinant 1 [3, p. 491. If 01is of orthogonal type, and A is trivial, we can detect the quotient G/T by determinant and spinor norm. The case when A is a matrix ring over a noncommutative division ring D is less well understood. If K is a local or global field, and 01has orthogonal type, then D must be a quaternion ring. The corresponding result here has been proved by M. Kneser using induction and an exceptional isomorphism.
    [Show full text]
  • Homomorphisms of Two-Dimensional Linear Groups Over a Ring of Stable Range One
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Algebra 303 (2006) 30–41 www.elsevier.com/locate/jalgebra Homomorphisms of two-dimensional linear groups over a ring of stable range one Carla Bardini a,∗,YuChenb a Dipartimento di Matematica, Università di Milano, Italy b Dipartimento di Matematica, Università di Torino, Italy Received 30 July 2004 Communicated by Efim Zelmanov Abstract Let R be a commutative domain of stable range 1 with 2 a unit. In this paper we describe the homo- morphisms between SL2(R) and GL2(K) where K is an algebraically closed field. We show that every non-trivial homomorphism can be decomposed uniquely as a product of an inner automorphism and a homomorphism induced by a morphism between R and K. We also describe the homomorphisms be- tween GL2(R) and GL2(K). Those homomorphisms are found of either extensions of homomorphisms from SL2(R) to GL2(K) or the products of inner automorphisms with certain group homomorphisms from GL2(R) to K. © 2006 Elsevier Inc. All rights reserved. Keywords: Minimal stable range; Two-dimensional linear group; Homomorphism Introduction Let R be a commutative domain of stable range 1 and K an algebraically closed field. The main purpose of this paper is to determinate the homomorphisms from SL2(R),aswellas GL2(R),toGL2(K) provided that 2 is a unit of R. The study of homomorphisms between linear groups started by Schreier and van der Waerden in [19] where they found that the automorphisms of a general linear group GLn over com- plex number field have a “standard” form, which is the composition of an inner automorphism, * Corresponding author.
    [Show full text]
  • Abelian Groups 1 Definition
    Abelian groups 1 Definition An Abelian group is a set A with a binary operation satisfying the following conditions: ◦ (A1) For all a;b;c A, we have a (b c) = (a b) c (the associative law). 2 ◦ ◦ ◦ ◦ (A2) There is an element e A such that a e = a for all a A. 2 ◦ 2 (A3) For any a A, there exists b A such that a b = e. 2 2 ◦ (A4) For all a;b A, we have a b = b a (the commutative law). 2 ◦ ◦ The first three axioms are almost identical to the axioms for a group, except that in the case of a group we normally say, for example, a 1 = 1 a = a in Axiom (A2) (this is obviously not necessary if (A4) holds). Thus,◦ an Abelian◦ group is a group satisfying the commutative law. Since many important Abelian groups arise as additive structures in various number systems (the integers, real numbers, integers mod m, etc.), it is very com- mon to write the group operation as + instead of ; with this convention it is natural to write 0 instead of e in (A2), and a instead◦ of b in (A3). We usually adopt this convention, which we describe as− “additive notation”. However, mul- tiplicative structures (the positive rationals, the non-zero complex numbers) also 1 give Abelian groups, and for these we might write ab for a b, 1 for e, and a− for b in (A3); we say that such a group is written with “multiplicative◦ notation”. A homomorphism is a map χ : A B between Abelian groups satisfying ! χ(a1 a2) = χ(a1) χ(a2).
    [Show full text]
  • Extremely Amenable Groups and Banach Representations
    Extremely Amenable Groups and Banach Representations A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University In partial fulfillment of the requirements for the degree Doctor of Philosophy Javier Ronquillo Rivera May 2018 © 2018 Javier Ronquillo Rivera. All Rights Reserved. 2 This dissertation titled Extremely Amenable Groups and Banach Representations by JAVIER RONQUILLO RIVERA has been approved for the Department of Mathematics and the College of Arts and Sciences by Vladimir Uspenskiy Professor of Mathematics Robert Frank Dean, College of Arts and Sciences 3 Abstract RONQUILLO RIVERA, JAVIER, Ph.D., May 2018, Mathematics Extremely Amenable Groups and Banach Representations (125 pp.) Director of Dissertation: Vladimir Uspenskiy A long-standing open problem in the theory of topological groups is as follows: [Glasner-Pestov problem] Let X be compact and Homeo(X) be endowed with the compact-open topology. If G ⊂ Homeo(X) is an abelian group, such that X has no G-fixed points, does G admit a non-trivial continuous character? In this dissertation we discuss some reformulations of this problem and its connections to other mathematical objects such as extremely amenable groups. When G is the closure of the group generated by a single map T ∈ Homeo(X) (with respect to the compact-open topology) and the action of G on X is minimal, the existence of non-trivial continuous characters of G is linked to the existence of equicontinuous factors of (X, T ). In this dissertation we present some connections between weakly mixing dynamical systems, continuous characters on groups, and the space of maximal chains of subcontinua of a given compact space.
    [Show full text]
  • Group Theory - QMII 2017/18
    Group theory - QMII 2017/18 Daniel Aloni 0 References 1. Lecture notes - Gilad Perez 2. Lie algebra in particle physics - H. Georgi 3. Google... 1Motivation As a warm up let us motivate the need for Group theory in physics. It comes out that Group theory and symmetries share very similar properties. This motivates us to learn more sophisticated tools of Group theory and how to implement them in modern physics. We will take rotations as an example but the following holds for any symmetry: 1. Arotationofarotationisalsoarotation-R = R R . 3 2 · 1 2. Multiplication of rotations is associative - R (R R )=(R R ) R . 3 · 2 · 1 3 · 2 · 1 3. The identity 1 is a unique rotation which leaves the system unchanged and commutes with all other rotations. 4. For each rotation we can rotate the system backwards. This inverse 1 1 rotation is unique and satisfies - R− R = R R− = 1. · · 1 This list is exactly the list of axioms that defines a group. A Group is a pair (G, )ofasetG and a product s.t. · · 1. Closure - g ,g G , g g G. 8 1 2 2 2 · 1 2 2. Associativity - g ,g ,g G,(g g ) g = g (g g ). 8 1 2 3 2 3 · 2 · 1 3 · 2 · 1 3. There is an identity e G,s.t. g G, e g = g e = g. 2 8 2 · · 1 1 4. Every element g G has an inverse element g− G,s.t. g g− = 2 2 · 1 g− g = e. · What we would like to learn? Here are few examples, considering the rotations again.
    [Show full text]
  • Linear Algebraic Groups
    Linear algebraic groups N. Perrin November 9, 2015 2 Contents 1 First definitions and properties 7 1.1 Algebraic groups . .7 1.1.1 Definitions . .7 1.1.2 Chevalley's Theorem . .7 1.1.3 Hopf algebras . .8 1.1.4 Examples . .8 1.2 First properties . 10 1.2.1 Connected components . 10 1.2.2 Image of a group homomorphism . 10 1.2.3 Subgroup generated by subvarieties . 11 1.3 Action on a variety . 12 1.3.1 Definition . 12 1.3.2 First properties . 12 1.3.3 Affine algebraic groups are linear . 14 2 Tangent spaces and Lie algebras 15 2.1 Derivations and tangent spaces . 15 2.1.1 Derivations . 15 2.1.2 Tangent spaces . 16 2.1.3 Distributions . 18 2.2 Lie algebra of an algebraic group . 18 2.2.1 Lie algebra . 18 2.2.2 Invariant derivations . 19 2.2.3 The distribution algebra . 20 2.2.4 Envelopping algebra . 22 2.2.5 Examples . 22 2.3 Derived action on a representation . 23 2.3.1 Derived action . 23 2.3.2 Stabilisor of the ideal of a closed subgroup . 24 2.3.3 Adjoint actions . 25 3 Semisimple and unipotent elements 29 3.1 Jordan decomposition . 29 3.1.1 Jordan decomposition in GL(V ).......................... 29 3.1.2 Jordan decomposition in G ............................. 30 3.2 Semisimple, unipotent and nilpotent elements . 31 3.3 Commutative groups . 32 3.3.1 Diagonalisable groups . 32 3 4 CONTENTS 3.3.2 Structure of commutative groups . 33 4 Diagonalisable groups and Tori 35 4.1 Structure theorem for diagonalisable groups .
    [Show full text]
  • Cosets and Lagrange's Theorem
    COSETS AND LAGRANGE'S THEOREM KEITH CONRAD 1. Introduction Pick an integer m 6= 0. For a 2 Z, the congruence class a mod m is the set of integers a + mk as k runs over Z. We can write this set as a + mZ. This can be thought of as a translated subgroup: start with the subgroup mZ and add a to it. This idea can be carried over from Z to any group at all, provided we distinguish between translation of a subgroup on the left and on the right. Definition 1.1. Let G be a group and H be a subgroup. For g 2 G, the sets gH = fgh : h 2 Hg; Hg = fhg : h 2 Hg are called, respectively, a left H-coset and a right H-coset. In other words, a coset is what we get when we take a subgroup and shift it (either on the left or on the right). The best way to think about cosets is that they are shifted subgroups, or translated subgroups. Note g lies in both gH and Hg, since g = ge = eg. Typically gH 6= Hg. When G is abelian, though, left and right cosets of a subgroup by a common element are the same thing. When an abelian group operation is written additively, an H-coset should be written as g + H, which is the same as H + g. Example 1.2. In the additive group Z, with subgroup mZ, the mZ-coset of a is a + mZ. This is just a congruence class modulo m.
    [Show full text]
  • Lecture Notes Group Theory
    Lecture Notes Group Theory Course held by Notes written by Prof. Kenichi Konishi Francesco Paolo Maiale Department of Mathematics Pisa University February 17, 2018 Disclaimer These are the notes I have written during the Group Theory course, held by Professor Kenichi Konishi in the first semester of the academic year 2017/2018. These include all the topics that were discussed during the lectures, but I took the liberty to extend the Mathematical side (e.g., the homotopy group chapter). To report any mistakes, misprints, or if you have any question, feel free to send me an email to francescopaolo (dot) maiale (at) gmail (dot) com. Acknowledgments I would like to thank the user Gonzalo Medina, from StackExchange, for the code (avail- able here) of the margin notes frame. Contents I Representation Theory7 1 Group Theory8 1.1 Set Theory.....................................8 1.2 Elementary Definitions and Basic Examples................... 10 1.2.1 Main Examples in Physics......................... 12 1.2.2 Back to Group Theory........................... 17 1.3 Finite Groups.................................... 21 2 Lie Groups and Lie Algebras 25 2.1 Definitions and Main Properties.......................... 25 2.1.1 Local Behavior: Lie Algebras....................... 26 2.1.2 Adjoint Representation.......................... 27 2.1.3 Examples.................................. 28 2.2 Lie Algebra..................................... 30 2.2.1 SubAlgebras................................ 31 2.3 Killing Form.................................... 32 2.3.1 Examples.................................. 33 2.4 Casimir Operator.................................. 34 3 The Fundamental Group π1(M) 36 3.1 Topological Spaces................................. 36 3.2 Homotopy...................................... 37 3.2.1 Path-Components............................. 38 3.2.2 Homotopy.................................. 39 3.3 The Fundamental Group.............................. 40 3.3.1 Path Homotopy..............................
    [Show full text]