DOCSLIB.ORG

1:34 Pm April 11, 2013 Red.Tex

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

1:34 p.m. April 11, 2013 Red.tex The structure of reductive groups Bill Casselman University of British Columbia, Vancouver [email protected] An algebraic group defined over F is an algebraic variety G with group operations specified in algebraic terms. For example, the group GLn is the subvariety of (n + 1) × (n + 1) matrices A 0 0 a with determinant det(A) a = 1. The matrix entries are well behaved functions on the group, here for 1 example a = det− (A). The formulas for matrix multiplication are certainly algebraic, and the inverse of a matrix A is its transpose adjoint times the inverse of its determinant, which are both algebraic. Formally, this means that we are given (a) an F •rational multiplication map G × G −→ G; (b) an F •rational inverse map G −→ G; (c) an identity element—i.e. an F •rational point of G. I’ll look only at affine algebraic groups (as opposed, say, to elliptic curves, which are projective varieties). In this case, the variety G is completely characterized by its affine ring AF [G], and the data above are respectively equivalent to the specification of (a’) an F •homomorphism AF [G] −→ AF [G] ⊗F AF [G]; (b’) an F •involution AF [G] −→ AF [G]; (c’) a distinguished homomorphism AF [G] −→ F . The first map expresses a coordinate in the product in terms of the coordinates of its terms. For example, in the case of GLn it takes xik −→ xij ⊗ xjk . j In addition, these data are subject to the group axioms. I’ll not say anything about the general theory of such groups, but I should say that in practice the specification of an algebraic group is often indirect—as a subgroup or quotient, say, of another simpler one. Contents 1. Algebraic tori 2. Split reductive groups 3. Automorphisms 4. The Bruhat decomposition 5. From root datum to group 6. Quasi•split reductive groups 7. Reductive groups over finite fields 8. Unramified groups over a p•adic field 9. References The standard references are [Steinberg:1968], [Humphreys:1975], [Borel:1991], and [Springer:1998]. The structure of reductive groups (1:34 p.m. April 11, 2013) 2 1. Algebraic tori The standard reference for the material in this section is §8 of [Borel:1991]. The multiplicative group over a field F is that of non•zero elements of F . It is isomorphic to the group 2 of points in F satisfying the equation xy = 1, and is hence an algebraic group Gm. Its affine ring is n the quotient F [x, y]/(xy − 1), which may be expressed as F [x, 1/x]. Similarly, the product (Gm) of n G ±1 ±1 copies of m has affine ring F [x1 ,...,xn ]. An algebraic torus defined over a field F is by definition an algebraic group defined over F that is n isomorphic to a product (Gm) after base extension to an algebraic closure of F . For example, the equation x2 + y2 =1 defines an algebraic group over R, and since x2 + y2 = (x + iy)(x − iy) a change of coordinates identifies the group over C with Gm. n SPLIT TORI. A torus defined over F that is isomorphic to some (Gm) over the field F itself is called an F •split torus. A character of a split torus T is an algebraic homomorphism from it to Gm, which means a ring ±1 ±1 ∗ homomorphism from F [x ] to F [xi ] compatible with group operations. Define X (T ) be the group of all characters of the split torus T . n n Suppose we have chosen an identification f of T with (Gm) . To k = (ki) in Z corresponds the character k ki ki x = (xi ) −→ xi . ∗ n ∗ This gives us a group homomorphism f from Z to X (T ). On the other hand, let X∗(T ) be the group n of algebraic homomorphisms from Gm to T —the covharacters of T . To each ℓ = (ℓi) in Z corresponds G ℓi Zn the map from m to T taking x to (xi ), and this gives us a map f∗ from to X∗(T ). ∗ [xstarT] 1.1. Proposition. The maps f∗ and f defined just above are isomorphisms. k Proof. Since the characters of Gm itself are all of the form x → x and (x1,...,xn) = (x1, 1,..., 1) · · · (1,..., 1, xn) , ∗ ki ki the map f is surjective. If xi = 1 for all (xi) then each x =1 for all x, hence every ki =0. Thus ∗ f is injective. The argument for X∗ is similar. [star-lin-ind] 1.2. Proposition. The characters of T form a linear basis of AF [T ]. × Proof. After choosing coordinates, embed X∗(T ) in Hom(T (F ), F ) and apply the following well known result: [lin-ind-chars] 1.3. Lemma. If G is a group and E a field, any set of distinct homomorphisms χ: G −→ E× are linearly independent over E. I postpone the proof to the end of this section. ∗ The two free Z•modules X (T ) and X∗(T ) are canonically dual to each other, which means that ∗ X (T ) = HomZ X∗(T ), Z . The structure of reductive groups (1:34 p.m. April 11, 2013) 3 ∨ ∗ ∨ Why is this? Given λ in X∗(T ) and µ in X (T ), the composite µ ◦ λ is a character og Gm, hence of the form x → xn for some unique n, which is defined a µ, λ∨. In short ∨ µ ◦ λ∨: x −→ xµ,λ . A split torus T is completely and simply characterized by X∗(T ). For one thing, we may describe the F • rational points of the torus as Hom(X∗(T ), F ×). For another, if we are given an algebraic homomorphism of tori ϕ: S → T , there exists a corresponding map ϕ∗: X∗(T ) → X∗(S). This is functorial. Conversely, ∗ ∗ ∗ since AF (T ) is the group algebra of X (T ), a homomorphism X (T ) → X (S) gives rise to an algebraic homomorphism. Thus the category of split algebraic tori is contravariantly equivalent to that of free Z•modules of finite rank. In particular the group of automorphisms of T may be canonically identified with GL X∗(T ) , which may in turn be non•canonically identified with GLn(Z). In other words, automorphisms of a split torus are of the form (x ) −→ xai,j i j j where ai,j is an integral matrix of determinant ±1. In particular all automorphisms are invariant under automorphisms of the field of definition, since the equations of the automorphism do not involve the field. The following is essentially a characterization of tori among Zariski•connected algebraic groups. [tori-reductive] 1.4. Proposition. Every F •rational algebraic representation of a split torus is a direct sum of characters. Proof. A representation of T is an algebraic homomorphism from T to GLn(F ). Because it is algebraic ±1 and the affine ring of T is isomorphic to F [xi ], we may write π(t)= cχ χ(t) , χ where each χ is a character of T and each cχ is an n × n matrix. Since π(1) = I we see that I = cχ , while since χ(st)= χ(s)χ(t) and the χ are linearly independent we see that cχ if χ = ρ cχcρ = 0 otherwise. n This implies that F is the direct sum of the images of the idempotent operators cχ, on which T acts as χ. CLASSIFICATION. How do we classify algebraic tori over an arbitrary field? We know that every torus becomes split over an algebraically closed extension of the base field. But one can do better: [split-separable-closure] 1.5. Theorem. Every torus defined over F splits over some finite Galois extension. I’ll give the proof at the end of this section. Say T splits over the a Galois extension E/F , with Galois group G. The endomorphism ring of a split torus of dimension n is isomorphic to Mn(Z) independently of the field of definition, and all automorphisms are Galois•invariant. The theory of Galois descent (see [Serre:1959]) therefore asserts that isomorphism classes of tori defined over F that split over E are in bijection with conjugacy classes of homomorphisms from G to GL X∗(T ) , each of which then gives rise in turn to an automorphism of The structure of reductive groups (1:34 p.m. April 11, 2013) 4 the E•rational points on the torus. If that associated to σ is ϕσ, then the group of F •rational points on σ the new torus is to be identified with the points x of T such that ϕσ(x)= x for all σ. Example. The real group x2 + y2 =1 becomes isomorphic to Gm upon base field extension to C. Its character group may be identified with Z, with conjugation acting as z → z−1, since (x + iy)−1 = x − iy. Example. Suppose E/F to be a separable quadratic extension. The multiplicative group E× may be identified with the group of F •rational points on a two•dimensional torus over F . This may be seen in two ways. • The group Gm over E is the group of (w,z) with wz =1. Let 1, α be an F •basis of E/F , with α2 − Bα + C =0 . The equations for Gm oevr E can be written as 1 = (u + vα)(x + yα) = ux + vyα2 + (uy + vx)α = ux + vy(Bα − C) + (uy + vx)α = (ux − Cvy) + (Bvy + uy + vx)α , which amount to the pair of equations ux − Cvy =1 Bvy + uy + vx =0 defining a two•dimensional group over F .
Recommended publications
  • Arxiv:1906.01274V1 [Math.NT] 4 Jun 2019 Rpsto .] Aqe Ishmefas Rvdsoepofi I Yale 1.4.1]

    Arxiv:1906.01274V1 [Math.NT] 4 Jun 2019 Rpsto .] Aqe Ishmefas Rvdsoepofi I Yale 1.4.1]

    CHOW’S THEOREM FOR SEMI-ABELIAN VARIETIES AND BOUNDS FOR SPLITTING FIELDS OF ALGEBRAIC TORI CHIA-FU YU Abstract. A theorem of Chow concerns homomorphisms of two abelian vari- eties under a primary field extension base change. In this paper we generalize Chow’s theorem to semi-abelian varieties. This contributes to different proofs of a well-known result that every algebraic torus splits over a finite separable field extension. We also obtain the best bound for the degrees of splitting fields of tori. 1. Introduction Let k be a field, k¯ an algebraic closure of k, and ks the separable closure of k in k¯. A connected algebraic k-group T is an algebraic torus if there is a k¯-isomorphism d T ⊗k k¯ ≃ (Gm) ⊗k k of algebraic groups for some integer d ≥ 0. We say T splits d over a field extension K of k if there is a K-isomorphism T ⊗k K ≃ (Gm) ⊗k K. This paper is motivated from the following fundamental result. Theorem 1.1. Any algebraic k-torus T splits over ks. In other words, T splits over a finite separable field extension of k. This theorem is well known and it is stated and proved in the literature several times. Surprisingly, different authors choose their favorite proofs which are all quite different. As far as we know, the first proof is given by Takashi Ono [40, Proposition 1.2.1]. Armand Borel gives a different proof in his book Linear Algebraic Groups; see [3, Proposition 8.11]. In the second edition of his book Linear Algebraic Groups [49], T.A.
  • Material on Algebraic and Lie Groups

    Material on Algebraic and Lie Groups

    2 Lie groups and algebraic groups. 2.1 Basic Definitions. In this subsection we will introduce the class of groups to be studied. We first recall that a Lie group is a group that is also a differentiable manifold 1 and multiplication (x, y xy) and inverse (x x ) are C1 maps. An algebraic group is a group7! that is also an algebraic7! variety such that multi- plication and inverse are morphisms. Before we can introduce our main characters we first consider GL(n, C) as an affi ne algebraic group. Here Mn(C) denotes the space of n n matrices and GL(n, C) = g Mn(C) det(g) =) . Now Mn(C) is given the structure nf2 2 j 6 g of affi ne space C with the coordinates xij for X = [xij] . This implies that GL(n, C) is Z-open and as a variety is isomorphic with the affi ne variety 1 Mn(C) det . This implies that (GL(n, C)) = C[xij, det ]. f g O Lemma 1 If G is an algebraic group over an algebraically closed field, F , then every point in G is smooth. Proof. Let Lg : G G be given by Lgx = gx. Then Lg is an isomorphism ! 1 1 of G as an algebraic variety (Lg = Lg ). Since isomorphisms preserve the set of smooth points we see that if x G is smooth so is every element of Gx = G. 2 Proposition 2 If G is an algebraic group over an algebraically closed field F then the Z-connected components Proof.
  • Algebraic Tori As Degenerations of Abelian Varieties

    Algebraic Tori As Degenerations of Abelian Varieties

    Mathematika 64 (2018) 303–329 c 2018 University College London doi:10.1112/S0025579317000420 ALGEBRAIC TORI AS DEGENERATIONS OF ABELIAN VARIETIES KAI-WEN LAN AND JUNECUE SUH Abstract. We first show that every algebraic torus over any field, not necessarily split, can be realized as the special fiber of a semi-abelian scheme whose generic fiber is an absolutely simple abelian variety. Then we investigate which algebraic tori can be thus obtained, when we require the generic fiber of the semi-abelian scheme to carry non-trivial endomorphism structures. Contents 1 Introduction 303 2 One-dimensional case 305 3 Descent data for tori 307 4 Theory of degeneration 308 5 Conditions for degenerations 310 6 Proof of main theorem 316 7 Non-trivial endomorphisms 319 Acknowledgements 329 References 329 §1. Introduction. We start with a well-known example, the Dwork family of elliptic curves X 3 C Y 3 C Z 3 − 3t XY Z D 0 1 t over the projective line PQ over Q, where is the affine coordinate, with the identity section given by the point at infinity TX V Y V ZUDT1 V −1 V 0U. It also doubles as the modular curve of level 3. It has bad, semistable reduction at t D 1 and the third roots of unity. The corresponding Neron´ model has the split torus as the identity component of the fiber at t D 1 (because the equation becomes XYZ D 0), but at t D 1 we get a non-split torus T that sits in the short exact sequence / / p Norm / / ; 1 T ResQ.
  • Algebraic Tori: a Computational Approach

    Algebraic Tori: a Computational Approach

    Western University Scholarship@Western Electronic Thesis and Dissertation Repository 8-8-2017 12:00 AM Algebraic Tori: A Computational Approach Armin Jamshidpey The University of Western Ontario Supervisor Prof. Nicole Lemire The University of Western Ontario Joint Supervisor Prof. Eric Schost The University of Western Ontario Graduate Program in Mathematics A thesis submitted in partial fulfillment of the equirr ements for the degree in Doctor of Philosophy © Armin Jamshidpey 2017 Follow this and additional works at: https://ir.lib.uwo.ca/etd Recommended Citation Jamshidpey, Armin, "Algebraic Tori: A Computational Approach" (2017). Electronic Thesis and Dissertation Repository. 4692. https://ir.lib.uwo.ca/etd/4692 This Dissertation/Thesis is brought to you for free and open access by Scholarship@Western. It has been accepted for inclusion in Electronic Thesis and Dissertation Repository by an authorized administrator of Scholarship@Western. For more information, please contact [email protected]. Abstract The rationality problem for algebraic tori is well known. It is known that any algebraic torus is unirational over its field of definition. The first purpose of this work is to solve rational- ity problem for 5 dimensional stably rational algebraic tori with an indecomposable character lattice. In order to do so, we have studied the associated character lattices of the mentioned algebraic tori. For each character lattice L, either we see the lattice as an associated lattice to a root system (of which rationality of its corresponding algebraic torus is known) or we find a reduced component of L so that we can relate rationality of the associated algebraic torus to lower dimensions.
  • Alexandre Grothendieck Eléments De Biographie

    Alexandre Grothendieck Eléments De Biographie

    Alexandre Grothendieck Eléments de biographie « Ce qui vient au monde pour ne rien troubler ne mérite ni égards ni patience ». René Char, Fureur et mystère Le 14 novembre 2014, les journaux, les radios, les télévisions du monde entier ont annoncé la mort, à Saint-Girons, en Ariège, d’un mathématicien âgé de 86 ans, au nom imprononçable. Des millions de personnes ont ainsi appris l’existence de cet inconnu au lendemain même de sa disparition. Quant à la communauté mathématique, elle a appris la nouvelle avec émotion, une émotion que Cédric Villani résume par ces mots : « Aucun mathématicien vivant n’était plus révéré de ses pairs que lui. » Pendant trente-cinq ans, tous les ans ou presque, en début d’année, j’ai tenu à dire à mes élèves quelques mots sur ce mathématicien, que je croyais retiré dans une bergerie de l’Hérault. C’est de lui que je vais vous parler, ce soir 1. Alexandre Grothendieck fut l’un des plus grands mathématiciens du 20 e siècle. L’irruption de cet autodidacte dans le monde mathématique, en 1949, ses contributions décisives en analyse fonctionnelle et en géométrie algébrique au cours de vingt années de création ininterrompue, sa rupture spectaculaire avec le monde mathématique, en 1970, et sa longue retraite, ont fait de lui un personnage mythique : Grothendieck fut un génie solitaire au destin romanesque, comme la science pure aime à en produire, de temps en temps, depuis trois millénaires : Evariste Galois, Albert Einstein, Srinivasa Ramanujan, Ettore Majorana, Alan Turing, Stephen Hawking, Grigori Perelman… Solitaire,
  • Fundamental Algebraic Geometry

    Fundamental Algebraic Geometry

    http://dx.doi.org/10.1090/surv/123 hematical Surveys and onographs olume 123 Fundamental Algebraic Geometry Grothendieck's FGA Explained Barbara Fantechi Lothar Gottsche Luc lllusie Steven L. Kleiman Nitin Nitsure AngeloVistoli American Mathematical Society U^VDED^ EDITORIAL COMMITTEE Jerry L. Bona Peter S. Landweber Michael G. Eastwood Michael P. Loss J. T. Stafford, Chair 2000 Mathematics Subject Classification. Primary 14-01, 14C20, 13D10, 14D15, 14K30, 18F10, 18D30. For additional information and updates on this book, visit www.ams.org/bookpages/surv-123 Library of Congress Cataloging-in-Publication Data Fundamental algebraic geometry : Grothendieck's FGA explained / Barbara Fantechi p. cm. — (Mathematical surveys and monographs, ISSN 0076-5376 ; v. 123) Includes bibliographical references and index. ISBN 0-8218-3541-6 (pbk. : acid-free paper) ISBN 0-8218-4245-5 (soft cover : acid-free paper) 1. Geometry, Algebraic. 2. Grothendieck groups. 3. Grothendieck categories. I Barbara, 1966- II. Mathematical surveys and monographs ; no. 123. QA564.F86 2005 516.3'5—dc22 2005053614 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA.
  • Some Finiteness Results for Groups of Automorphisms of Manifolds 3

    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.
  • Automorphism Groups of Locally Compact Reductive Groups

    Automorphism Groups of Locally Compact Reductive Groups

    PROCEEDINGS OF THE AMERICAN MATHEMATICALSOCIETY Volume 106, Number 2, June 1989 AUTOMORPHISM GROUPS OF LOCALLY COMPACT REDUCTIVE GROUPS T. S. WU (Communicated by David G Ebin) Dedicated to Mr. Chu. Ming-Lun on his seventieth birthday Abstract. A topological group G is reductive if every continuous finite di- mensional G-module is semi-simple. We study the structure of those locally compact reductive groups which are the extension of their identity components by compact groups. We then study the automorphism groups of such groups in connection with the groups of inner automorphisms. Proposition. Let G be a locally compact reductive group such that G/Gq is compact. Then /(Go) is dense in Aq(G) . Let G be a locally compact topological group. Let A(G) be the group of all bi-continuous automorphisms of G. Then A(G) has the natural topology (the so-called Birkhoff topology or ^-topology [1, 3, 4]), so that it becomes a topo- logical group. We shall always adopt such topology in the following discussion. When G is compact, it is well known that the identity component A0(G) of A(G) is the group of all inner automorphisms induced by elements from the identity component G0 of G, i.e., A0(G) = I(G0). This fact is very useful in the study of the structure of locally compact groups. On the other hand, it is also well known that when G is a semi-simple Lie group with finitely many connected components A0(G) - I(G0). The latter fact had been generalized to more general groups ([3]).
  • LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2

    LIE GROUPS and ALGEBRAS NOTES Contents 1. Definitions 2

    LIE GROUPS AND ALGEBRAS NOTES STANISLAV ATANASOV Contents 1. Definitions 2 1.1. Root systems, Weyl groups and Weyl chambers3 1.2. Cartan matrices and Dynkin diagrams4 1.3. Weights 5 1.4. Lie group and Lie algebra correspondence5 2. Basic results about Lie algebras7 2.1. General 7 2.2. Root system 7 2.3. Classification of semisimple Lie algebras8 3. Highest weight modules9 3.1. Universal enveloping algebra9 3.2. Weights and maximal vectors9 4. Compact Lie groups 10 4.1. Peter-Weyl theorem 10 4.2. Maximal tori 11 4.3. Symmetric spaces 11 4.4. Compact Lie algebras 12 4.5. Weyl's theorem 12 5. Semisimple Lie groups 13 5.1. Semisimple Lie algebras 13 5.2. Parabolic subalgebras. 14 5.3. Semisimple Lie groups 14 6. Reductive Lie groups 16 6.1. Reductive Lie algebras 16 6.2. Definition of reductive Lie group 16 6.3. Decompositions 18 6.4. The structure of M = ZK (a0) 18 6.5. Parabolic Subgroups 19 7. Functional analysis on Lie groups 21 7.1. Decomposition of the Haar measure 21 7.2. Reductive groups and parabolic subgroups 21 7.3. Weyl integration formula 22 8. Linear algebraic groups and their representation theory 23 8.1. Linear algebraic groups 23 8.2. Reductive and semisimple groups 24 8.3. Parabolic and Borel subgroups 25 8.4. Decompositions 27 Date: October, 2018. These notes compile results from multiple sources, mostly [1,2]. All mistakes are mine. 1 2 STANISLAV ATANASOV 1. Definitions Let g be a Lie algebra over algebraically closed field F of characteristic 0.
  • Underlying Varieties and Group Structures 3

    Underlying Varieties and Group Structures 3

    UNDERLYING VARIETIES AND GROUP STRUCTURES VLADIMIR L. POPOV Abstract. Starting with exploration of the possibility to present the underlying variety of an affine algebraic group in the form of a product of some algebraic varieties, we then explore the naturally arising problem as to what extent the group variety of an algebraic group determines its group structure. 1. Introduction. This paper arose from a short preprint [16] and is its extensive extension. As starting point of [14] served the question of B. Kunyavsky [10] about the validity of the statement, which is for- mulated below as Corollary of Theorem 1. This statement concerns the possibility to present the underlying variety of a connected reductive algebraic group in the form of a product of some special algebraic va- rieties. Sections 2, 3 make up the content of [16], where the possibility of such presentations is explored. For some of them, in Theorem 1 is proved their existence, and in Theorems 2–5, on the contrary, their non-existence. Theorem 1 shows that there are non-isomorphic reductive groups whose underlying varieties are isomorphic. In Sections 4–10, we explore the problem, naturally arising in connection with this, as to what ex- tent the underlying variety of an algebraic group determines its group structure. In Theorems 6–8, it is shown that some group properties (dimension of unipotent radical, reductivity, solvability, unipotency, arXiv:2105.12861v1 [math.AG] 26 May 2021 toroidality in the sense of Rosenlicht) are equivalent to certain geomet- ric properties of the underlying group variety. Theorem 8 generalizes to solvable groups M.
  • ALGEBRAIC GROUPS: PART III Contents 10. the Lie Algebra of An

    ALGEBRAIC GROUPS: PART III Contents 10. the Lie Algebra of An

    ALGEBRAIC GROUPS: PART III EYAL Z. GOREN, MCGILL UNIVERSITY Contents 10. The Lie algebra of an algebraic group 47 10.1. Derivations 47 10.2. The tangent space 47 10.2.1. An intrinsic algebraic definition 47 10.2.2. A naive non-intrinsic geometric definition 48 10.2.3. Via point derivations 49 10.3. Regular points 49 10.4. Left invariant derivations 51 10.5. Subgroups and Lie subalgebras 54 10.6. Examples 55 10.6.1. The additive group Ga. 55 10.6.2. The multiplicative group Gm 55 10.6.3. The general linear group GLn 56 10.6.4. Subgroups of GLn 57 10.6.5. A useful observation 58 10.6.6. Products 58 10.6.7. Tori 58 10.7. The adjoint representation 58 10.7.1. ad - The differential of Ad. 60 Date: Winter 2011. 46 ALGEBRAIC GROUPS: PART III 47 10. The Lie algebra of an algebraic group 10.1. Derivations. Let R be a commutative ring, A an R-algebra and M an A-module. A typical situation for us would be the case where R is an algebraically closed field, A the ring of regular functions of an affine k-variety and M is either A itself, or A=M, where M is a maximal ideal. Returning to the general case, define D, an M-valued R-derivation of A, to be a function D : A ! M; such that D is R-linear and D(ab) = a · D(b) + b · D(a): We have used the dot here to stress the module operation: a 2 A and D(b) 2 M and a · D(b) denotes the action of an element of the ring A on an element of the module M.
  • Geometric Representation Theory of SL2(R)

    Geometric Representation Theory of SL2(R)

    Geometric representation theory of SL2(R) Justin Campbell April 20, 2015 1 Introduction The representation theory of SL2(R) is of interest to number theorists and physicists, notably through its connection to the theory of modular forms. The importance of the discrete series representations where these modular forms live was recogized classically, but a full classification of the irreducible admissible representations was only accomplished by Bargmann in 1947. The goal of this note is to give an algebro- geometric account of the classification using the theory of D-modules. Notations: GR = SL2(R) G = SL2(C) g = sl2(C) U(g) = the universal enveloping algebra of g Z(g) = the center of U(g) ∼ 1 KR = SO2(R) = S ∼ K = SO2(C) = Gm 2 Harish-Chandra modules We are interested a priori in representations of GR acting on certain topological vector spaces (e.g. Hilbert, Banach, and Fr´echet spaces), but this involves functional analysis. If we na¨ıvely attempted to classify irreducible admissible representations up to isomorphism, we would find that many representations which are defined by the same formulas and therefore \the same" representation-theoretically are non-isomorphic for topological reasons. Example 2.1. Consider the adjoint action of G on 1 =∼ S1. This is induces an action of G on the Hilbert R PR R space L2(S1), the Banach spaces Lp(S1) for all p, the Frechet space C1(S1), etc. For this reason we will formulate the notion of infinitesimal equivalence, which is insensitive to these analytic subtleties. This leads naturally to the consideration of Harish-Chandra modules.