Sato, hiroshi, journal of mathematics of kyoto university, 1971. Raghunathan received june 19, 1978 the paper referred to in the title appeared in this journal in 1966 vol. Explicit formulas for discrete series characters on noncompact cartan subgroups are given. Discrete subgroups of real semisimple lie groups g a margulison some groups of motions of noncompact nonsingular symmetric spaces of rank 1 m e novodvorskistructure of topological locally projectivelynilpotent groups, and of groups with a normalizer condition v p platonovrecent citations lower bound for the volumes of quaternionic hyperbolic. Since the notion of lie group is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e. In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show. For these groups, we have seen how to split them into a direct product of a compact and a vector group, how to complement the commutator group by an abelian lie group, and that all compact lie groups are linear. Harishchandra has defined the schwartz space, vg, on g. Sacksteder, an extension of the minkowskihlawka theorem, mathematika 56 2010, 203216, essential use was made of the fact that in its natural linear action the real symplectic group, spn. By lie groups we not only mean real lie groups, but also the sets of krational points of algebraic groups over local fields k and their direct products. The global classification of noncompact real semisimple lie groups can be carried out in a similar but more complicated way.
Pdf volume invariant and maximal representations of discrete. Folge a series of modern surveys in mathematics 17. Maximal operators associated to discrete subgroups. Mar 16, 2006 the present book is devoted to lattices, i. Factorizations in the irreducible characters of compact semisimple lie groups andrew rupinski alexandre kirillov, advisor our goal is to describe factorizations of the characters of irreducible representations of compact semisimple lie groups.
Mapping class groups of surfaces with the actions on the teichmuller spaces. A geometric construction of the discrete series for. Harmonic analysis on some types of semisimple lie groups eguchi, masaaki, proceedings of the japan academy, 1974. An equivalent definition of a simple lie group follows from the lie correspondence. Mostow notes by gopal prasad no part of this book may be reproduced in any form by print, micro. Discrete subgroups of semisimple lie groups ergebnisse. A literature reference for finiteness of the center of a connected complexanalytic lie group with semisimple lie algebra i assume this is your definition of semisimple for the analytic group is ch. Discrete subgroups of lie groups and discrete transformation. On orispherical subgroups of a semisimple lie group maruyama, shigeya, kodai mathematical seminar reports, 1968 representations of complex semisimple lie groups and lie algebras parthasarathy, k. A key property of a lie group is that a curved space can be studied, using. Geometry of biinvariant subsets of complex semisimple lie groups. Margulis, discrete subgroups of real semisimple lie groups. Does every nonabelian lie group have a finite subgroup.
Then r acts without fixed points on the left on the symmetric space x gk, and can therefore be identified with the fundamental. Chevalley group, and from them most of the finite simple groups can be obtained with the exception of the alternating group and the 26 sporadic groups, cf. Kupers our goal today is to compare the cohomology of bgand bg, for ga lie group. For a thorough treatment of the history of the subject, see. Harmonic analysis of tempered distributions on semisimple lie groups of real rank one james g. They prove that a connected, semisimple lie group contains a dense copy of any finitely generated, fully residually free group a limit group in selas terminology. In mathematics, a simple lie group is a connected nonabelian lie group g which does not have nontrivial connected normal subgroups. Chapter i presents a brief resume, with occasional indications of proofs, of the theory of semisimple lie groups up to but not including cartans highest weight theory for finitedimensional representations and the theory of parabolic subgroups. On the first cohomology of discrete subgroups of semi. You might also be interested in a theorem of breuillard, gelander, souto and storm. Find materials for this course in the pages linked along the left. Iii the structure of semisimple lie algebras and algebraic groups in char. This thesis is an expository account of three central theorems in the representation theory of semisimple lie groups, namely the theorems of borelweilbott, casselmanosborne and kostant.
Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. In this video we talk about the history of the search for simple groups, the role of symmetry in tesselations, both euclidean, spherical and hyperbolic, and. The group gis called reductive or semisimple if g has the corresponding property. These formulas are used to give a simple formula for the fourier transform of orbital integrals of regular. Knapp offers a survey of representation theory of semisimple lie groups in a way that reflects the spirit of the subject and corresponds to the natural learning process. A note on the multipliers and projective representations of semisimple lie groups by bhaskar bagchi and gadadhar misra indian statistical institute, bangalore summary. Finite simple subgroups of semisimple complex lie groups. Although theorems are always stated precisely, many illustrative examples or classes of. Unfortunately there are some connected lie groups, such as nilpotent lie groups. A similar classification holds for compact real semisimple lie groups, each of which is imbedded in a unique complex semisimple lie group as a maximal compact subgroup see lie group, compact. All lattices in a nilpotent lie group are uniform, and if is a connected simply connected nilpotent lie group equivalently it does not contain a nontrivial compact subgroup then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup this generalises the fact that a discrete subgroup in a vector space is a lattice if and only if it spans the vector space.
Readings introduction to lie groups mathematics mit. Dimensional gap in semisimple compact lie groups via. A geometric construction of the discrete series for semisimple lie groups 3 k local integrability of the harishchandra characters. Of these i follows from the fact that h appears as a subrepresentation of an induced representation see the simple proof by casselman 9. On the first cohomology of discrete subgroups of semisimple. Up to the middle of the 20th century one basically studied individual classes of discrete subgroups of lie groups occurring in arithmetic, function theory and physics.
Marsden abstractwe consider the optimal control of mechanical systems on lie groups and develop numerical methods which exploit the structure of the state space and preserve the system motion invariants. The discrete series of semisimple groups peter hochs september 5, 2019 abstract these notes contain some basic facts about discrete series representations of semisimple lie groups. A major achievement in the theory of discrete subgroups of semisimple lie groups is margulis superrigidity theorem. Lattices in semisimple lie groups a theorem of wang. This book is a model of exposition and an invaluable resource for both graduate students and researchers. The fundamental result is formulated in the introduction.
These formulas are used to give a simple formula for the fourier transform of orbital integrals of regular semisimple orbits. Harmonic inner automorphisms of compact connected semisimple lie groups park, joonsik, tohoku mathematical journal, 1990. Representations of certain solvable lie groups on hilbert. An important technical point is that a simple lie group may contain discrete normal subgroups, hence being a simple lie group is different from being simple as an abstract group.
Discrete series for semisimple lie groups i construction of ibivariant eigendistributio1ns by harishchandra the institute for advanced study, princeton, n. A detailed treatment of the geometric aspects of discrete groups was carried out by raghunathan in his book discrete subgroups of lie groups which. The links to ca, gt, ft, and ag in the pdf file will work if the files are placed. Dynamics in the study of discrete subgroups of lie groups. In the preceding chapter, we studied groups with a compact lie algebra. For a large part, they summarise relevant material from knapps book 12. Some transitive linear actions of real simple lie groups laura geatti and martin moskowitz abstract. In this paper we show that strong generalizations of the measurable livsic theorem for cocycles taking values in connected noncompact linear semisimple lie groups, a canonical example being sl2. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semisimple lie groups. Our results can be applied to the theory of algebraic groups over global fields. A siegel domain of type ii is a complex domain of the.
Everything is much better if you additionally require that your lie groups are compact. The simple lie groups with holomorphic discrete series are those whose symmetric space is hermitian. Volume invariant and maximal representations of discrete subgroups of lie groups. Discrete geometric optimal control on lie groups marin kobilarov and jerrold e. Discrete subgroups of semisimple lie groups gregori a. Harmonic analysis of tempered distributions on semisimple. If one of two commensurable subgroups of a locally compact topological group is a discrete subgroup or a lattice, or a uniform discrete subgroup, so is the other. Simple groups, lie groups, and the search for symmetry i. However, in the present case of noncompact real forms of complex simple lie algebras, the maximal compact subgroups of the associated simple real lie groups are not necessarily semisimple. Ix t 7 dco o oe d represents the contribution of the discrete series to the plancherel formula of g, we intend to obtain explicit formulas. With the goal of describing simple lie groups, we analyze semisimple complex lie algebras by their root systems to.
This necessitates a description of the second cohomology groups of homogeneous spaces of lie. Besides discrete subgroups of lie groups, two other very important discrete transformation groups are. On the characters of a semisimple lie group harishchandra, bulletin of the american mathematical society. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. Dimensional gap in semisimple compact lie groups via fourier. Livsics theorem for semisimple lie groups article pdf available in ergodic theory and dynamical systems 2105. The rst of these realizes all the irreducible holomorphic representations of a complex semisimple lie group gin the cohomology of certain sheaves of. For example, if we consider the classical lie groups on, son, sln etc, it is easy to find finiteorder elements considering the diagonal matrices with diagonal elements taken. Together with the commutative lie group of the real numbers, r \displaystyle \mathbb r, and that of the unitmagnitude complex numbers, u 1 the unit circle, simple lie groups give the atomic blocks. Algebraic groups, lie groups, and their arithmetic subgroups. Discrete subgroups of semisimple lie group researchgate. Harmonic analysis of tempered distributions on semisimple lie. Let g and b be the lie algebras of g and b respectively.
Introduction and statement of main theorem the purpose of this paper is to prove a maximal theorem for averages taken over suitable discrete subvarieties of nilpotent lie groups. Holomorphic discrete series representations are the easiest discrete series. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. Introduction we are concerned in this article with the study of representations of lie groups g on hilbert spaces of holomorphic functions on a ghomogeneous siegel domain of type ii. Holomorphic discrete series representation wikipedia. With the goal of describing simple lie groups, we analyze semisimple complex lie algebras by their root systems to classify simple lie algebras. Breuillard, gelander, souto, storm, dense embeddings of surface groups. Oneparameter subgroups and a lie subgroup of an infinite dimensional rotation group. Let be the free group on generators, the outer automorphism group is the most important group in combinatorial group theory. Our interest, by and large, is in a special class of discrete subgroups of lie groups, viz. In mathematics, a holomorphic discrete series representation is a discrete series representation of a semisimple lie group that can be represented in a natural way as a hilbert space of holomorphic functions. Suppose further that g is linear and that r contains no elements of finite order. Unfortunately theorem 1 of the paper as it stands is incorrect. This paper introduces lie groups and their associated lie algebras.
Maximal operators associated to discrete subgroups of nilpotent lie groups by akos magyar, elias m. The classification of simple complex lie algebras joshua bosshardt abstract. We assume a background in linear algebra, di erential manifolds, and covering spaces. On the representation theory of semisimple lie groups. Let g be a topological group and f an arbitrary group. Raghunathan and others published discrete subgroups of lie groups find, read and cite all the research you need on researchgate.
In mathematics, a simple lie group is a connected nonabelian lie group g which does not have nontrivial connected normal subgroups together with the commutative lie group of the real numbers, and that of the unitmagnitude complex numbers, u1 the unit circle, simple lie groups give the atomic blocks that make up all finitedimensional connected lie groups via the operation of. It is assumed that the reader has considerable familiarity with lie groups and algebraic groups. Every compact connected lie group of positive dimension has a nontrivial maximal torus which has many elements of finite order in the noncompact connected case were still fine if the maximal compact subgroup is interesting. Here bg denotes the classifying space of gas a topological group, while bg denotes its classifying space as a discrete group. It seems to me that each nonabelian lie group has at least one actually many finiteorder elements or, in other terms, one or more finite subgroups. Some transitive linear actions of real simple lie groups. A lie algebra is reductive if and only if it is the direct sum of an abelian and a semisimple lie algebra. Discrete shapes can be described and analyzed using lie groups, which are mathematical structures having both algebraic and geometrical properties. We show that, for any connected semisimple lie group g, there is a natural isomorphism between the galois cohomology h2g,t with respect to the trivial action of g. A tempered distribution on g is a continuous linear functional on rg. These structures, borrowed from mathematical physics, are both algebraic groups and smooth manifolds. Polycyclic groups and arithmeticity of lattices in solvable lie groups. In particular, semisimple lie algebras are reductive. The positive characteristic case salehi golsefidy, alireza, duke mathematical journal, 2012.
556 1584 1529 1220 212 1089 106 389 1385 1613 783 1667 1484 668 22 1685 1575 154 548 1544 1017 1644 1208 1499 614 844 412 16 288 324 842 1411 179 639 410 148 60