The representation theory of complex semisimple lie groups has a long history beginning in 1950 with the fundamental work. 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. It is assumed that the reader has considerable familiarity with lie groups and algebraic groups. We give a method for constructing dense and free subgroups in real lie groups. Get your kindle here, or download a free kindle reading app. 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. Fix a borel subalgebra b and a cartan subalgebra h of g contained in b. Vinberg and others published discrete subgroups of lie groups find, read.
If gis a connected lie group then kg j eg can be calculated very easily from the representation theory of the maximal compact subgroup. Lectures on representations of complex semisimple lie groups. 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. Invariant eigendistributions on semisimple lie groups harishchandra, bulletin of the american mathematical society, 1963. Discrete subgroups of semisimple lie groups by gregori a. Lectures on discrete subgroups of lie groups download link. The rigidity theorem for ergodic actions of semi simple lie groups of rrank at least 2 27 shows in one. Discrete series for semisimple lie groups i construction of ibivariant eigendistributio1ns by harishchandra the institute for advanced study, princeton, n. On the representation theory of semisimple lie groups.
In this lecture, g always denotes a connected, noncompact, semisimple lie group. The theory is particularly rich for lattices in semisimple lie groups or more. On the characters of a semisimple lie group harishchandra, bulletin of the american mathematical society, 1955. If g is a connected noncompact semisimple lie group with. Let r be a discrete uniform subgroup of a connected simply connected. Margulis, 9783642057212, available at book depository with free delivery worldwide. Our interest, by and large, is in a special class of discrete subgroups of lie groups, viz. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. These formulas are used to give a simple formula for the fourier transform of orbital integrals of regular. Lectures on discrete subgroups of lie groups free book at ebooks directory. In the special case of subgroups of r n, this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all. However, i do not believe one can use this technique to construct simple discrete subgroups of lie groups.
Find materials for this course in the pages linked along the left. You can read online discrete subgroups of semisimple lie groups here in pdf, epub, mobi or docx formats. The point of this paper is to study the low dimensional cohomology theory of ergodic actions of semisimple lie groups and their lattice subgroups. Volume invariant and maximal representations of discrete subgroups of lie groups authors. Discrete subgroups of semisimple lie groups gregori a. Note that s0 is linear algebraic, because it is simple and center free.
Let g be a semisimple connected linear lie group, w, a finitedimensional irreducible representation of g, tr2 an infinitedimensional irreducible representation of g which has a nontrivial extension with w. Jun 02, 2014 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 introduction of continuous groups. 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. Alternatively, the groups can sometimes be treated via other categorizations, e. Algebraic groups, lie groups, and their arithmetic subgroups.
In this chapter, unless expressly stated otherwise, by lie algebra we mean a complex lie algebra. Volume invariant and maximal representations of discrete subgroups of lie groups. Pdf volume invariant and maximal representations of discrete. In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show. Besides discrete subgroups of lie groups, two other very important discrete transformation groups are. For a general reference on discrete subgroups of lie groups, see 14. Oneparameter subgroups and a lie subgroup of an infinite dimensional rotation group. The global classification of noncompact real semi simple lie groups can be carried out in a similar but more complicated way. Polycyclic groups and arithmeticity of lattices in solvable lie groups.
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. For a large part, they summarise relevant material from knapps book 12. Discrete geometric optimal control on lie groups marin kobilarov and jerrold e. By looking at these examples i realized what i should have known long time ago, namely that infinite permutation group with dynkin diagram infinite line with integer nodes is virtually simple. Discrete subgroups of lie groups and discrete transformation. Next come the results which are crucial to our approach.
Readings introduction to lie groups mathematics mit. For every connected semisimple lie group s the adjoint representation ad has the property that ads is linear algebraic and the kernel which is the. Mostow notes by gopal prasad no part of this book may be reproduced in any form by print, micro. Harishchandraschwartzs algebras associated with discrete subgroups of semisimple lie groups. Other readers will always be interested in your opinion of the books youve read. Mapping class groups of surfaces with the actions on the teichmuller spaces.
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. The resulting formulas 52 and 55 are applicable for the case that among the irreducible representations of the given group there is at least one representation with a simple spectrum in particular, for all the. In lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete. Bieberbach reduced the study of fundamental groups of flat manifolds to the study of the discrete subgroup of the isometry group of r n, which was later developed into a more general theory about. Classifying space for proper actions and ktheory of group.
Holomorphic discrete series representation wikipedia. In particular we show that any dense subgroup of a connected semisimple real lie group g contains a free group on two generators which is still dense in g, and that any finitely generated dense subgroup in a connected nonsolvable lie group h contains a dense free subgroup of rank lie algebras of the free associative algebra and the free lie algebra morita, shigeyuki, sakasai, takuya, and suzuki, masaaki, duke mathematical journal, 20. Simple groups, lie groups, and the search for symmetry i. Realizing a lie group as an algebraic group is the. Download product flyer download highresolution cover. Since every real lie algebra can be complexified, most of our results also have immediate consequences for real lie algebras. The subcase of the noncompact case of complex lie groups is of interest due to the special simplicity of the theory for these groups and the unusual parallels which exist between this theory and the theory for the compact case. Sungwoon kim, inkang kim submitted on 22 may 2012 v1, last revised 21 sep 2012 this version, v2. The global classification of noncompact real semisimple lie groups can be carried out in a similar but more complicated way. The importance of these cohomology groups arises from the role they play in deformation theory. Discrete subgroups of lie groups pdf free download epdf. Whether youve loved the book or not, if you give your honest and detailed thoughts then people will find new books that are right for them.
Every reductive algebraic group is the quotient of a product of a torus and a semisimple group by a. It is shown how s is essentially determined by f, using the point of view of group extensions. Folge a series of modern surveys in mathematics on free. A simple method is developed for computing the eigenvalues of the invariant operators the socalled casimir operators c p of arbitrary order p for semisimple lie groups. This is possible because the complexification of a real semisimple lie algebra is also semisimple see 1. Harmonic analysis of tempered distributions on semisimple lie. A lattice in a lie group is usually defined as a discrete subgroup with finite covolume. Pdf download discrete subgroups of semisimple lie groups free. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semisimple lie groups. Unfortunately theorem 1 of the paper as it stands is incorrect. A tempered distribution on g is a continuous linear functional on rg.
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. Folge a series of modern surveys in mathematics on. A note on the multipliers and projective representations of semisimple lie groups by bhaskar bagchi and gadadhar misra indian statistical institute, bangalore summary. Let g and b be the lie algebras of g and b respectively. Folge a series of modern surveys in mathematics on free shipping on qualified orders. Indecomposable representations of semisimple lie groups by birgit speh abstract. At present, only the split case is covered in chapter i, only the semisimple case is covered in detail in chapter ii, and only a partial summary of. Volume invariant and maximal representations of discrete. Discrete subgroups of lie groups the interest in discrete. Raghunathan received june 19, 1978 the paper referred to in the title appeared in this journal in 1966 vol. In lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. 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. Casimir operators for semisimple lie groups iopscience. Does every nonabelian lie group have a finite subgroup.
Discrete subgroups of semisimple lie groups book, 1991. A geometric construction of the discrete series for semisimple lie groups 3 k local integrability of the harishchandra characters. Harishchandra has defined the schwartz space, vg, on g. On the first cohomology of discrete subgroups of semisimple. Dynamics in the study of discrete subgroups of lie groups. A detailed treatment of the geometric aspects of discrete groups was carried out by raghunathan in his book discrete subgroups of lie groups which appeared in 1972. Harishchandra schwartzs algebras associated with discrete subgroups of semisimple lie groups. Let be the free group on generators, the outer automorphism group is the most important group in combinatorial group theory. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi simple lie groups. On orispherical subgroups of a semisimple lie group maruyama, shigeya, kodai mathematical seminar reports, 1968. Aim of the course after nishing the course the students should 1 understand the algebraic structure of some fundamental classes of lie groups such as nilpotent and semisimple lie groups, 2 master the basic techniques in classi cation of re ection groups and root. We discuss the discrete groups that arise in this way in an appendix. The author not only proves results in the classical geometry read more. The following concepts provide a possible formal way to isolate two precise classes of groups corresponding to those two extremes.
Let f be a discrete uniform subgroup of a connected simply connected solvable lie group s. 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. Lie group if and only if it contains a subgroup of finite index which is torsionfree and of. The first of these realizes all the irreducible holomorphic representations of a complex semisimple lie group g in the cohomology of certain sheaves of. We show that certain discrete subgroups of semisimple lie groups satisfy rigidity properties and that a subclass of these discrete groups are actually of finite covolume. Lectures on locally symmetric spaces and arithmetic groups. Example of a discrete uniform subgroup of a lie group which. On some rigid subgroups of semisimple lie groups springerlink. The simple lie groups with holomorphic discrete series are those whose symmetric space is hermitian.
Semisimple lie groups and discrete subgroups by robert j. This book is a model of exposition and an invaluable resource for both graduate students and researchers. Discrete versions of g a connected lie group may have discrete subgroups that. Harishchandra discrete series for semisimple lie groups. Advances in mathematics 15, 112123 1975 discrete subgroups of lie groups g. Advances in mathematics 41, 781 1981 vanishing theorems for lie algebra cohomology and the cohomology of discrete subgroups of semisimple lie groups wilfried schmid department of mathematics, harvard university, cambridge, massachusetts 028 1. Our study of real semisimple lie groups and algebras is based on the theory of complex semisimple lie groups developed in ch. On measure rigidity of unipotent subgroups of 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.
Harmonic analysis on semisimple groups modern treatment. Since the semisimple part s of the lie group g does not have compact. Vanishing theorems for lie algebra cohomology and the. The emphasis in these lectures is on the semisimple case. 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. Arithmeticity of discrete subgroups containing horospherical lattices yves benoist and s ebastien miquel abstract let g be a semisimple real algebraic lie group of real rank at least two and u the unipotent radical of a nontrivial parabolic subgroup.
The terminology introduced above is coherent with this, as a theorem due to borel and harishchandra states that an arithmetic subgroup in a semisimple lie group is of finite covolume the discreteness is obvious. Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. If gis torsion free and discrete then they are the khomology groups of the classifying space bg. Thus a consists precisely of the weights which are dominant with respect to the system of positive roots a.
Discrete subgroups of semisimple lie groups ergebnisse. A similar classification holds for compact real semi simple lie groups, each of which is imbedded in a unique complex semi simple lie group as a maximal compact subgroup see lie group, compact. Lie algebra gl k for some k, or a semisimple lie algebra. We prove that the harishchandraschwartz space associated with a discrete subgroup of a. Let g be a lie subgroup of the lie group of all automorphisms of a 11. Of these i follows from the fact that h appears as a subrepresentation of an induced representation see the simple proof by casselman 9.
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. To do this we shall say that a sequence of lattices converges. A geometric construction of the discrete series for. Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple lie algebra. Harmonic analysis of tempered distributions on semisimple lie groups of real rank one james g. Mostow department of mathematics, yale university, new haven, connecticut 06520 dedicated to garrett birkhoff the interest in discrete subgroups of lie groups has arisen from two seemingly separate sourcesone geometric and one arithmetic. Harishchandraschwartzs algebras associated with discrete. On framed simple lie groups minami, haruo, journal of the mathematical society of japan, 2016. Our results can be applied to the theory of algebraic groups over global fields.
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