Let 2 be a locally compact group, with a closed, unimodular, cocompact subgroup h. Discrete subgroups of lie groups and discrete transformation. Let be the free group on generators, the outer automorphism group is the most important group in combinatorial group theory. Finiteness results for lattices in certain lie groups greenleaf, frederick p. Discrete subgroups of solvable lie groups have been fairly thoroughly studied, but the results are less complete than those obtained for nilpotent groups. Warner, foundations of differentiable manifolds and lie groups djvu. If we consider the special case ho identify, then we are dealing with a discrete group h. However, i do not believe one can use this technique to construct simple discrete subgroups of lie groups.
Get your kindle here, or download a free kindle reading app. A major achievement in the theory of discrete subgroups of semisimple lie groups is margulis superrigidity theorem. Discrete subgroups of lie groups the interest in discrete. 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. Finite simple subgroups of semisimple complex lie groups. Margulis, 9783642057212, available at book depository with free delivery worldwide. Tata institute of fundamental research, bombay 1969. 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. Shahshahani received july 7, 1971 introduction let g be a semisimple lie group acting as a group of linear transformations on the vector space v, k a maximal compact subgroup of g. If h is a lie group, and h,rad h e, then h is unimodular.
Let g be a connected semisimple lie group and r a discrete subgroup such that the quotient gr is compact. This content was uploaded by our users and we assume good faith they have the permission to share this book. Harmonic analysis on semisimple lie groups harishchandra, bulletin of the american mathematical society, 1970. 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. Blocks of normal subgroups, automorphisms of groups, and the alperinmckay conjecture murai, masafumi, kyoto journal of mathematics, 2014. Continuous cohomology, discrete subgroups, and representations of reductive groups.
The first one of the three embeddings is the most natural one. Then r acts without fixed points on the left on the symmetric space x gk, and can therefore be identified with the fundamental. Zimmer, infinitesimal rigidity for smooth actions of discrete subgroups of lie groups, j. Readings introduction to lie groups mathematics mit. In this article those discrete subgroups of the group g of real unimodular matrices of order three are investigated which have the property that the factor space of the group g by them has finite volume and is not compact. Discrete subgroups of lie groups and applications to moduli. 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.
Check our section of free ebooks and guides on lie algebra now. For a large part, they summarise relevant material from knapps book 12. On the first cohomology of discrete subgroups of semisimple. Harishchandraschwartzs algebras associated with discrete subgroups of semisimple lie groups.
Our interest, by and large, is in a special class of discrete subgroups of lie groups, viz. Besides discrete subgroups of lie groups, two other very important discrete transformation groups are. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semisimple lie groups. A geometric construction of the discrete series for semisimple lie groups 3 k local integrability of the harishchandra characters. For a real lie algebra, negative definite killing form equivalent to semisimple and compact already has the group connected piece at least compact. Find materials for this course in the pages linked along the left.
Dynamics in the study of discrete subgroups of lie groups. Closedness of connected semisimple lie subgroups of. Readers should be familiar with differential manifolds and the elementary theory of lie groups and lie algebras. Volume 7 of tata institute of fundamental research studies in mathematics tata institute of fundamental research volume 7 of studies in mathematics. The reductive groups are the ones that arise in practice, and we shall see in this section that they differ from semisimple groups only trivially. Zimmer,on the cohomology of ergodic actions of semisimple lie groups and discrete subgroups, am. The book by borel and wallach is a classic treatment of the use of cohomology in representation theory, particularly in the setting of automorphic forms and discrete subgroups. It turns out that if g is semisimple then no generality is lost in. The modular group is a discrete subgroup of since is discrete in. Let g and b be the lie algebras of g and b respectively. Wallach, seminar on the cohomology of discrete subgroups of semisimple groups, to appear. The final chapter proves the basic theorems on maximal compact subgroups of lie groups. Let po be a finite dimensional representation of g.
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. Suppose further that g is linear and that r contains no elements of finite order. Donde surgen las sombras david lozano pdf, geeraar jaciel pdf995. A geometric construction of the discrete series for semisimple lie groups. On the first cohomology of discrete subgroups of semisimple lie. A detailed treatment of the geometric aspects of discrete groups was carried out by raghunathan in his book discrete subgroups of lie groups which. Discrete mathematics elliptic curves fourier analysis functional analysis fractals. Variants of kazhdans property for subgroups of semisimple. Hubsch submitted on 30 mar 2010 v1, last revised 17 jun 2017 this version, v2. Fuchsian groups are, by definition, discrete subgroups of the isometry group of the hyperbolic plane. A geometric construction of the discrete series for. 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. Its aim is to present a detailed ac count of some of the recent work on the geometric aspects of the theory of discrete subgroups of lie groups.
Mapping class groups of surfaces with the actions on the teichmuller spaces. Any lattice in a solvable lie group is a uniform discrete subgroup. 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. The problem of classifying the real reductive groups largely reduces to classifying the simple lie groups. Jump to navigation jump to search nolan russell wallach. 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. Covering semisimple groups by subgroups 665 8 the gap group, gap groups, algorithms, and programming, version.
Discrete subgroups of semisimple lie groups by gregori a. Harishchandraschwartzs algebras associated with discrete. On orispherical subgroups of a semisimple lie group maruyama, shigeya, kodai mathematical seminar reports, 1968. Cocompact subgroups of semisimple lie groups lemma 1. Whittaker functions on semisimple lie groups hashizume, michihiko, hiroshima mathematical journal, 1982. Of these i follows from the fact that h appears as a subrepresentation of an induced representation see the simple proof by casselman 9. Mostow notes by gopal prasad no part of this book may be reproduced in any form by print, micro. The chapters concerning discrete subgroups of semi simple lie groups are essentially concerned with results which were obtained in the 1960s. In particular, every connected semisimple lie group meaning that its lie algebra is semisimple is reductive. Also, the lie group r is reductive in this sense, since it can be viewed as the identity component of gl1,r. Invariant eigendistributions on semisimple lie groups harishchandra, bulletin of the american mathematical society, 1963. Then is a lie group since it is the inverse image of the regular value 1 under or, it is a closed subgroup of a lie group. For semisimple groups themselves, our first examples were sl2. Our results can be applied to the theory of algebraic groups over global fields.
A fuchsian group that preserves orientation and acts on the upper halfplane model of the hyperbolic plane is a discrete subgroup of the lie group psl2, r, the group of orientation preserving isometries of the upper halfplane model of the. Discrete subgroups of semisimple lie groups gregori a. A duality operation in the character ring of a finite group of lie type curtis, c. Characters of averaged discrete series on semisimple real lie groups. Mar 16, 2006 the present book is devoted to lattices, i. Basic material on affine connections and on locally or globally riemannian and hermitian symmetric spaces is covered. It is assumed that the reader has considerable familiarity with lie groups and algebraic groups. In 1978 he was an invited speaker with talk the spectrum of compact quotients of semisimple lie groups. Then 2 is unimodular, and hh has a finite z invariant measure.
Discrete subgroups of lie groups pdf free download epdf. Firstly, is a lattice, namely is finite, but is infinite. Free lie algebra books download ebooks online textbooks. Continuous cohomology, discrete subgroups and representations of reductive groups, annals of mathematical studies 94, 1980, 2nd edition. Contemporary mathematics volume cocompact subgroups of.
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