It introduces the reader to the representation theory of compact lie groups. Models of representations of compact lie groups springerlink. Sections 417 deal with the general theory of lie groups. We have chosen a geometrical and analytical approach since we feel that this is the easiest way to motivate and establish the theory and to indicate relations to other branches of mathematics. The general theory is welldeveloped in case g is a locally compact hausdorff topological group and the representations are strongly continuous the theory has been widely applied in quantum mechanics since the 1920s. This result certainly fails for non compact groups. Learning the irreducible representations of commutative lie groups scribes a representation of such a group, and show how it can be learned from pairs of images related by arbitrary and unobserved transformations in the group. Peter, bulletin new series of the american mathematical society, 1989. Representation theory of classical compact lie groups. Lie 18421899, who rst encountered lie groups and the associated lie algebras when studying symmetries of partial di erential equations. Certain types of lie groupsnotably, compact lie groupshave the property that every finitedimensional representation is isomorphic to a direct sum of irreducible representations. Learning the irreducible representations of commutative lie. Su n, the group of unitary complex matrices, with lie algebrasu n consisting of skew hermitian matrices, and son, the group of orthogonal real matrices with lie algebra son consisting of antisymmetric matrices.
Galois representations and elliptic curves 3 from a representation g. Learning the irreducible representations of commutative. Ametricapproachto representationsofcompactliegroups. A lattice is a discrete additive subgroup l v such that the set l spans the vector space v over r. This very important special chapter of the representation theory of compact lie groups is key to further. In mathematics, a unitary representation of a group g is a linear representation. Langlands parameters and finitedimensional representations. Ramanan no part of this book may be reproduced in any form by print, micro. Inside a general semisimple lie group there is a maximal compact subgroup, and the representation theory of such groups, developed largely by harishchandra, uses intensively the restriction of a representation to such a subgroup, and also the model of weyls character theory. They can therefore be identi ed with the tate algebras khpnu 1pnu dias kbanach spaces.
For each compact lie algebra g and each real representationv of g we construct a twostep nilpotent lie groupng, v, endowed with a natural leftinvariant riemannian metric. Signature quantization and representations of compact lie groups victor guillemin and etienne rassart department of mathematics, massachusetts institute of technology, cambridge, ma 029 contributed by victor guillemin, may 11, 2004 we discuss some applications of signature quantization to the representation theory of compact lie groups. The representationring of a compact lie group numdam. Fulton and harris, representation theory, a first course bump, lie groups hall, lie. The complete reducibility to finite dimensional representations result you read about is a consequence of or part of, depending on how you phrase it the peterweyl theorem. There arent very many geometrically flavored books on lie groups. For noncompact lie groups and various choices of spaces of maps, one needs.
An approach through invariants and representations universitext. Representations of compact lie groups theodor brocker, tammo. Induced representations of compact groups 56 chapter 4. We summarize the results of this chapter in the peterweyltheorem. Gelfand pairs attached to representations of compact lie. In such cases, the classification of representations reduces to the classification of irreducible representations. In chapter 5 we study complex representations of connected abelian lie groups tori. Theodor brocker and tammo tom dieck, representations of compact lie groups. Lie groups and representations of locally compact groups by f. Lie algebras, though mentioned occasionally, are not used in an essential way.
I general remarks in this talk a loop group lg will mean the group of smooth maps from the circle s i to a compact lie group g. Why all irreducible representations of compact groups are. Representation theory of compact groups and complex. Lie groups beyond an introduction representations of compact. This introduction to the representation theory of compact lie groups follows. The t i are referred to as generators of the lie group.
Lgroups and bk david vogan introduction compact lie finite chevalley compact padic old reasons for listening to langlands gln everybodys favorite reductive grouplocal f. The general theory is welldeveloped in case g is a locally compact hausdorff topological group and the representations are strongly continuous. The baumconnes conjecture and parametrization of representations. Representations of compact padic analytic groups 457 the subalgebras ou n. Lie groups beyond an introduction representations of.
These lectures lead by a relatively straight path from the end of a onesemester course in lie groups through the langlands classi. Assuming no prior knowledge of lie groups, this book covers the structure and representation theory of compact lie groups. C, embeddings of free groups as schottky groups, or embeddings of uniform lattices. Contraction of compact semisimple lie groups via berezin quantization cahen, benjamin, illinois journal of. The study of the continuous unitary representations of g leads to that of the irreducible representations.
If youre perfectly happy with a more algebraic treatment, read no further. The representations of compact groups are particularly well behaved, which we shall show in chapter 4. Many years ago i wrote the book lie groups, lie algebras, and some of their applications ny. I can think of only this one and compact lie groups by sepanski. Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016. Lie groups beyond an introduction, with emphasis on chapters iv, v, and ix. A more general assertion holds for compact topological groups, without the assumption of lieness, due to the compactness of hilbertschmidt operators. Signature quantization and representations of compact lie groups. Lectures on lie groups and representations of locally compact. Models of representations of compact lie groups, preprint ipm no. In the rst problem set, one exercise will be to prove frobenius reciprocity in the lie algebra case, and in the lie group case, for compact.
A morphism of the associated representations of lie algebras is the same as a morphism. Apart from the intrinsic interest, the theory of lie groups and their representations is used in various parts of mathematics. Theodor brocker and tammo tom dieck, representations of compact lie groups find. Buy compact lie groups graduate texts in mathematics on. Brocker and tom dieck representations of compact lie groups o some other books on lie groups i have not looked carefully at all of these books myself, pdf files of some of them are available for download through pitt library. Relations between representations of compact lie groups and complex lie algebras 63 4. A metric approach to representations of compact lie groups. Introduction to representations theory of lie groups. Theyre all conjugate inside g, so it doesnt matter which one we choose. Representations of compact lie groups graduate texts in. Most lie groups books fall into one of two categories.
This very important special chapter of the representation theory of compact lie groups is key to further study of the representations of nonabelian lie groups. Theodor brocker and tammo tom dieck, representations. Gelfand pairs attached to representations of compact lie groups. Lie groups and compact groups lie groups and compact. R, quasifuchsian representations of surface groups into psl2. We shall study lie groups and lie algebras, and representations of compact lie groups.
Over the course of the years i realized that more than 90% of the most useful material in that book could be presented in less than 10% of the space. We may therefore view them as rigid analytic quantizations. Representations of su2 and related groups introduction this appendix provides background material on manifolds, vector bundles, and lie groups, which are used throughout the book. We present some classical results on compact lie groups, such as the peterweyl theorem, on. Three other nice references are the springer graduate texts in mathematics representations of compact lie groups by brocker and tom dieck, representation theory by fulton and harris, and introduction to lie algebras and representation theory by humphreys. From this, weyl was able to determine the irreducible representations of compact groups. We have rigidly adhered to the analytic approach in establishing the relations between lie groups and lie algebras. Teg is called the adjoint representation of g in teg.
Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. Chapter 15 representations of compact groups sciencedirect. The chapter discusses complete reducibility, irreducible representations of a compact group, characters of compact groups, representations of finite groups, and use of compact subgroups of arbitrary groups. Signature quantization and representations of compact lie. This book is based on several courses given by the authors since 1966. Raghunathan and others published lie groups and algebraic groups find, read and cite all the. Glv of a padic lie group, where v is a vector space over q p, we get a representation of the lie algebra of g, denoted lie g. Indeed, we will give a full classification of the manifoldsng, v which are commutative. This is one of the main reasons we will mostly be restricting our attention to representations of compact lie.
We discuss integration on a lie group, the lie algebra, and the exponential map from the lie algebra to the lie group. We apply these objects to general results on representations. W depending on the sort of group one is dealing with, note that one additionally has to specify what class of maps one is dealing with. Sun, the group of unitary complex matrices, with lie algebrasun consisting of skew hermitian matrices. A topological group is a topological space g with a group structure such that the multiplication map m. Proofs, if any, are either sketched or given in simpler cases. One reason for study ing such groups is that they are the simplest examples of infinite dimensional lie groups. Representation theory of compact groups and complex reductive. Translated from funktsionalnyi analiz i ego prilozheniya, vol.
Baumconnes for lie groups here the conjecture is proved. Blending algebra, analysis, and topology, the study of compact lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general lie groups. One of the key ideas in the theory of lie groups is to replace the global object, the group, with its local or linearized version, which lie himself called its infinitesimal group and which has since become known as its lie algebra. On irreducible representations of compact padic analytic. Anosov representations into lie groups of rank one are exactly convex cocompact representations see theorem1. Contraction of compact semisimple lie groups via berezin quantization cahen, benjamin, illinois journal of mathematics, 2009 on the large time behavior of heat kernels on lie groups lohoue, noel and alexopoulos, georgios, duke mathematical journal, 2003. Compact lie groups and representation theory lecture notes. Representations of compact lie groups springerlink. I warmly thank lotte hollands for providing me with latex files for these. G connected semisimple group k maximal compact subgroup assume kng has a gequivariant spincstructure, given by a kequivariant clifford algebra representation c.
Glv of a padic lie group, where v is a vector space over q p, we get a representation of the lie algebra of g, denoted lieg. Lie groups form a class of topological groups, and the compact lie groups have a particularly welldeveloped theory. A more general assertion holds for compact topological groups, without the assumption of lie ness, due to the compactness of hilbertschmidt operators. Compact commutative groups are also known as toroidal groups, so we refer to this model as toroidal subgroup analysis. Representations of compact lie groups pdf free download epdf. The main goal of this paper is to show that this construction produces many new gelfand pairs associated with nilpotent lie groups. Themainreferencesusedwere8forbanachalgebratheory,17forthespectral theorem and its application to schurs lemma, and 5 for locally compact groupsandrepresentationtheory. Lectures on lie groups and representations of locally. The f ijk are referred to as the structure constants of the lie group. Lie groups, lie algebras, and their representations representations of. Induced representations and frobenius reciprocity math g4344, spring 2012. Tammo tom dieck, transformation groups and representation theory may, j. Direct sums of representations and complete reducibility 79 6.
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