Finite Maximal Tori
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Affine Springer Fibers and Affine Deligne-Lusztig Varieties
Affine Springer Fibers and Affine Deligne-Lusztig Varieties Ulrich G¨ortz Abstract. We give a survey on the notion of affine Grassmannian, on affine Springer fibers and the purity conjecture of Goresky, Kottwitz, and MacPher- son, and on affine Deligne-Lusztig varieties and results about their dimensions in the hyperspecial and Iwahori cases. Mathematics Subject Classification (2000). 22E67; 20G25; 14G35. Keywords. Affine Grassmannian; affine Springer fibers; affine Deligne-Lusztig varieties. 1. Introduction These notes are based on the lectures I gave at the Workshop on Affine Flag Man- ifolds and Principal Bundles which took place in Berlin in September 2008. There are three chapters, corresponding to the main topics of the course. The first one is the construction of the affine Grassmannian and the affine flag variety, which are the ambient spaces of the varieties considered afterwards. In the following chapter we look at affine Springer fibers. They were first investigated in 1988 by Kazhdan and Lusztig [41], and played a prominent role in the recent work about the “fun- damental lemma”, culminating in the proof of the latter by Ngˆo. See Section 3.8. Finally, we study affine Deligne-Lusztig varieties, a “σ-linear variant” of affine Springer fibers over fields of positive characteristic, σ denoting the Frobenius au- tomorphism. The term “affine Deligne-Lusztig variety” was coined by Rapoport who first considered the variety structure on these sets. The sets themselves appear implicitly already much earlier in the study of twisted orbital integrals. We remark that the term “affine” in both cases is not related to the varieties in question being affine, but rather refers to the fact that these are notions defined in the context of an affine root system. -
Faithful Abelian Groups of Infinite Rank Ulrich Albrecht
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 103, Number 1, May 1988 FAITHFUL ABELIAN GROUPS OF INFINITE RANK ULRICH ALBRECHT (Communicated by Bhama Srinivasan) ABSTRACT. Let B be a subgroup of an abelian group G such that G/B is isomorphic to a direct sum of copies of an abelian group A. For B to be a direct summand of G, it is necessary that G be generated by B and all homomorphic images of A in G. However, if the functor Hom(A, —) preserves direct sums of copies of A, then this condition is sufficient too if and only if M ®e(A) A is nonzero for all nonzero right ¿ï(A)-modules M. Several examples and related results are given. 1. Introduction. There are only very few criteria for the splitting of exact sequences of torsion-free abelian groups. The most widely used of these was given by Baer in 1937 [F, Proposition 86.5]: If G is a pure subgroup of a torsion-free abelian group G, such that G/C is homogeneous completely decomposable of type f, and all elements of G\C are of type r, then G is a direct summand of G. Because of its numerous applications, many attempts have been made to extend the last result to situations in which G/C is not completely decomposable. Arnold and Lady succeeded in 1975 in the case that G is torsion-free of finite rank. Before we can state their result, we introduce some additional notation: Suppose that A and G are abelian groups. -
Notes 9: Eli Cartan's Theorem on Maximal Tori
Notes 9: Eli Cartan’s theorem on maximal tori. Version 1.00 — still with misprints, hopefully fewer Tori Let T be a torus of dimension n. This means that there is an isomorphism T S1 S1 ... S1. Recall that the Lie algebra Lie T is trivial and that the exponential × map× is× a group homomorphism, so there is an exact sequence of Lie groups exp 0 N Lie T T T 1 where the kernel NT of expT is a discrete subgroup Lie T called the integral lattice of T . The isomorphism T S1 S1 ... S1 induces an isomorphism Lie T Rn under which the exponential map× takes× the× form 2πit1 2πit2 2πitn exp(t1,...,tn)=(e ,e ,...,e ). Irreducible characters. Any irreducible representation V of T is one dimensio- nal and hence it is given by a multiplicative character; that is a group homomorphism 1 χ: T Aut(V )=C∗. It takes values in the unit circle S — the circle being the → only compact and connected subgroup of C∗ — hence we may regard χ as a Lie group map χ: T S1. → The character χ has a derivative at the unit which is a map θ = d χ :LieT e → Lie S1 of Lie algebras. The two Lie algebras being trivial and Lie S1 being of di- 1 mension one, this is just a linear functional on Lie T . The tangent space TeS TeC∗ = ⊆ C equals the imaginary axis iR, which we often will identify with R. The derivative θ fits into the commutative diagram exp 0 NT Lie T T 1 θ N θ χ | T exp 1 1 0 NS1 Lie S S 1 The linear functional θ is not any linear functional, the values it takes on the discrete 1 subgroup NT are all in N 1 . -
An Overview of Topological Groups: Yesterday, Today, Tomorrow
axioms Editorial An Overview of Topological Groups: Yesterday, Today, Tomorrow Sidney A. Morris 1,2 1 Faculty of Science and Technology, Federation University Australia, Victoria 3353, Australia; [email protected]; Tel.: +61-41-7771178 2 Department of Mathematics and Statistics, La Trobe University, Bundoora, Victoria 3086, Australia Academic Editor: Humberto Bustince Received: 18 April 2016; Accepted: 20 April 2016; Published: 5 May 2016 It was in 1969 that I began my graduate studies on topological group theory and I often dived into one of the following five books. My favourite book “Abstract Harmonic Analysis” [1] by Ed Hewitt and Ken Ross contains both a proof of the Pontryagin-van Kampen Duality Theorem for locally compact abelian groups and the structure theory of locally compact abelian groups. Walter Rudin’s book “Fourier Analysis on Groups” [2] includes an elegant proof of the Pontryagin-van Kampen Duality Theorem. Much gentler than these is “Introduction to Topological Groups” [3] by Taqdir Husain which has an introduction to topological group theory, Haar measure, the Peter-Weyl Theorem and Duality Theory. Of course the book “Topological Groups” [4] by Lev Semyonovich Pontryagin himself was a tour de force for its time. P. S. Aleksandrov, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko described this book in glowing terms: “This book belongs to that rare category of mathematical works that can truly be called classical - books which retain their significance for decades and exert a formative influence on the scientific outlook of whole generations of mathematicians”. The final book I mention from my graduate studies days is “Topological Transformation Groups” [5] by Deane Montgomery and Leo Zippin which contains a solution of Hilbert’s fifth problem as well as a structure theory for locally compact non-abelian groups. -
Algebraic Number Theory
Algebraic Number Theory Sergey Shpectorov January{March, 2010 This course in on algebraic number theory. This means studying problems from number theory with methods from abstract algebra. For a long time the main motivation behind the development of algebraic number theory was the Fermat Last Theorem. Proven in 1995 by Wiles with the help of Taylor, this theorem states that there are no positive integers x, y and z satisfying the equation xn + yn = zn; where n ≥ 3 is an integer. The proof of this statement for the particular case n = 4 goes back to Fibonacci, who lived four hundred years before Fermat. Modulo Fibonacci's result, Fermat Last Theorem needs to be proven only for the cases where n = p is an odd prime. By the end of the course we will hopefully see, as an application of our theory, how to prove the Fermat Last Theorem for the so-called regular primes. The idea of this belongs to Kummer, although we will, of course, use more modern notation and methods. Another accepted definition of algebraic number theory is that it studies the so-called number fields, which are the finite extensions of the field of ra- tional numbers Q. We mention right away, however, that most of this theory applies also in the second important case, known as the case of function fields. For example, finite extensions of the field of complex rational functions C(x) are function fields. We will stress the similarities and differences between the two types of fields, as appropriate. Finite extensions of Q are algebraic, and this ties algebraic number the- ory with Galois theory, which is an important prerequisite for us. -
Orders on Computable Torsion-Free Abelian Groups
Orders on Computable Torsion-Free Abelian Groups Asher M. Kach (Joint Work with Karen Lange and Reed Solomon) University of Chicago 12th Asian Logic Conference Victoria University of Wellington December 2011 Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 1 / 24 Outline 1 Classical Algebra Background 2 Computing a Basis 3 Computing an Order With A Basis Without A Basis 4 Open Questions Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 2 / 24 Torsion-Free Abelian Groups Remark Disclaimer: Hereout, the word group will always refer to a countable torsion-free abelian group. The words computable group will always refer to a (fixed) computable presentation. Definition A group G = (G : +; 0) is torsion-free if non-zero multiples of non-zero elements are non-zero, i.e., if (8x 2 G)(8n 2 !)[x 6= 0 ^ n 6= 0 =) nx 6= 0] : Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 3 / 24 Rank Theorem A countable abelian group is torsion-free if and only if it is a subgroup ! of Q . Definition The rank of a countable torsion-free abelian group G is the least κ cardinal κ such that G is a subgroup of Q . Asher M. Kach (U of C) Orders on Computable TFAGs ALC 2011 4 / 24 Example The subgroup H of Q ⊕ Q (viewed as having generators b1 and b2) b1+b2 generated by b1, b2, and 2 b1+b2 So elements of H look like β1b1 + β2b2 + α 2 for β1; β2; α 2 Z. -
Math 210C. Simple Factors 1. Introduction Let G Be a Nontrivial Connected Compact Lie Group, and Let T ⊂ G Be a Maximal Torus
Math 210C. Simple factors 1. Introduction Let G be a nontrivial connected compact Lie group, and let T ⊂ G be a maximal torus. 0 Assume ZG is finite (so G = G , and the converse holds by Exercise 4(ii) in HW9). Define Φ = Φ(G; T ) and V = X(T )Q, so (V; Φ) is a nonzero root system. By the handout \Irreducible components of root systems", (V; Φ) is uniquely a direct sum of irreducible components f(Vi; Φi)g in the following sense. There exists a unique collection of nonzero Q-subspaces Vi ⊂ V such that for Φi := Φ \ Vi the following hold: each pair ` (Vi; Φi) is an irreducible root system, ⊕Vi = V , and Φi = Φ. Our aim is to use the unique irreducible decomposition of the root system (which involves the choice of T , unique up to G-conjugation) and some Exercises in HW9 to prove: Theorem 1.1. Let fGjgj2J be the set of minimal non-trivial connected closed normal sub- groups of G. (i) The set J is finite, the Gj's pairwise commute, and the multiplication homomorphism Y Gj ! G is an isogeny. Q (ii) If ZG = 1 or π1(G) = 1 then Gi ! G is an isomorphism (so ZGj = 1 for all j or π1(Gj) = 1 for all j respectively). 0 (iii) Each connected closed normal subgroup N ⊂ G has finite center, and J 7! GJ0 = hGjij2J0 is a bijection between the set of subsets of J and the set of connected closed normal subgroups of G. (iv) The set fGjg is in natural bijection with the set fΦig via j 7! i(j) defined by the condition T \ Gj = Ti(j). -
Strongly Homogeneous Torsion Free Abelian Groups of Finite Rank
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 56, April 1976 STRONGLY HOMOGENEOUS TORSION FREE ABELIAN GROUPS OF FINITE RANK Abstract. An abelian group is strongly homogeneous if for any two pure rank 1 subgroups there is an automorphism sending one onto the other. Finite rank torsion free strongly homogeneous groups are characterized as the tensor product of certain subrings of algebraic number fields with finite direct sums of isomorphic subgroups of Q, the additive group of rationals. If G is a finite direct sum of finite rank torsion free strongly homogeneous groups, then any two decompositions of G into a direct sum of indecompos- able subgroups are equivalent. D. K. Harrison, in an unpublished note, defined a p-special group to be a strongly homogeneous group such that G/pG » Z/pZ for some prime p and qG = G for all primes q =£ p and characterized these groups as the additive groups of certain valuation rings in algebraic number fields. Rich- man [7] provided a global version of this result. Call G special if G is strongly homogeneous, G/pG = 0 or Z/pZ for all primes p, and G contains a pure rank 1 subgroup isomorphic to a subring of Q. Special groups are then characterized as additive subgroups of the intersection of certain valuation rings in an algebraic number field (also see Murley [5]). Strongly homoge- neous groups of rank 2 are characterized in [2]. All of the above-mentioned characterizations can be derived from the more general (notation and terminology are as in Fuchs [3]): Theorem 1. -
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]). -
Loop Groups and Twisted K-Theory III
Annals of Mathematics 174 (2011), 947{1007 http://dx.doi.org/10.4007/annals.2011.174.2.5 Loop groups and twisted K-theory III By Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman Abstract In this paper, we identify the Ad-equivariant twisted K-theory of a compact Lie group G with the \Verlinde group" of isomorphism classes of admissible representations of its loop groups. Our identification pre- serves natural module structures over the representation ring R(G) and a natural duality pairing. Two earlier papers in the series covered founda- tions of twisted equivariant K-theory, introduced distinguished families of Dirac operators and discussed the special case of connected groups with free π1. Here, we recall the earlier material as needed to make the paper self-contained. Going further, we discuss the relation to semi-infinite co- homology, the fusion product of conformal field theory, the r^oleof energy and a topological Peter-Weyl theorem. Introduction Let G be a compact Lie group and LG the space of smooth maps S1 !G, the loop group of G. The latter has a distinguished class of irreducible pro- jective unitary representations, the integrable lowest-weight representations. They are rigid; in fact, if we fix the central extension LGτ of LG there is a finite number of isomorphism classes. Our main theorem identifies the free abelian group Rτ (LG) they generate with a twisted version of the equivari- ant topological K-theory group KG(G), where G acts on itself by conjugation. The twisting is constructed from the central extension of the loop group. -
Covering Group Theory for Compact Groups
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Pure and Applied Algebra 161 (2001) 255–267 www.elsevier.com/locate/jpaa Covering group theory for compact groups Valera Berestovskiia, Conrad Plautb;∗ aDepartment of Mathematics, Omsk State University, Pr. Mira 55A, Omsk 77, 644077, Russia bDepartment of Mathematics, University of Tennessee, Knoxville, TN 37919, USA Received 27 May 1999; received in revised form 27 March 2000 Communicated by A. Carboni Abstract We present a covering group theory (with a generalized notion of cover) for the category of compact connected topological groups. Every such group G has a universal covering epimorphism 2 K : G → G in this category. The abelian topological group 1 (G):=ker is a new invariant for compact connected groups distinct from the traditional fundamental group. The paper also describes a more general categorial framework for covering group theory, and includes some examples and open problems. c 2001 Elsevier Science B.V. All rights reserved. MSC: Primary: 22C05; 22B05; secondary: 22KXX 1. Introduction and main results In [1] we developed a covering group theory and generalized notion of fundamental group that discards such traditional notions as local arcwise connectivity and local simple connectivity. The theory applies to a large category of “coverable” topological groups that includes, for example, all metrizable, connected, locally connected groups and even some totally disconnected groups. However, for locally compact groups, being coverable is equivalent to being arcwise connected [2]. Therefore, many connected compact groups (e.g. solenoids or the character group of ZN ) are not coverable. -
Contents 1 Root Systems
Stefan Dawydiak February 19, 2021 Marginalia about roots These notes are an attempt to maintain a overview collection of facts about and relationships between some situations in which root systems and root data appear. They also serve to track some common identifications and choices. The references include some helpful lecture notes with more examples. The author of these notes learned this material from courses taught by Zinovy Reichstein, Joel Kam- nitzer, James Arthur, and Florian Herzig, as well as many student talks, and lecture notes by Ivan Loseu. These notes are simply collected marginalia for those references. Any errors introduced, especially of viewpoint, are the author's own. The author of these notes would be grateful for their communication to [email protected]. Contents 1 Root systems 1 1.1 Root space decomposition . .2 1.2 Roots, coroots, and reflections . .3 1.2.1 Abstract root systems . .7 1.2.2 Coroots, fundamental weights and Cartan matrices . .7 1.2.3 Roots vs weights . .9 1.2.4 Roots at the group level . .9 1.3 The Weyl group . 10 1.3.1 Weyl Chambers . 11 1.3.2 The Weyl group as a subquotient for compact Lie groups . 13 1.3.3 The Weyl group as a subquotient for noncompact Lie groups . 13 2 Root data 16 2.1 Root data . 16 2.2 The Langlands dual group . 17 2.3 The flag variety . 18 2.3.1 Bruhat decomposition revisited . 18 2.3.2 Schubert cells . 19 3 Adelic groups 20 3.1 Weyl sets . 20 References 21 1 Root systems The following examples are taken mostly from [8] where they are stated without most of the calculations.