Wavefront sets of representations of lie groups pdf

These lecture notes were created using material from prof. We shall show that every irreducible representation of the group at least in. The term representation of a group is also used in a more general sense to mean any description of a group as a group of transformations of some mathematical object. Mat 4451196 introduction to representation theory chapter 1 representation theory of groups algebraic foundations.

They act on the set of tensors of a given rank, which is indeed a linear vector space. Chapter ii lie groups and lie algebras a lie group is, roughly speaking, an analytic manifold with a group structure such that the group operations are analytic. Contraction of irreducible tempered representations afgoustidis, alexandre, duke mathematical journal, 2020. This lie algebra is a quite fundamental object, that crops up at. B has lie algebra b, b is a closed connected subgroup of g, and b is its own normalizer in g. The topic of this course is not \ representation theory of reductive lie groups, but simply \ representation theory, or simply the study of lie groups. For these reasons, we eschew the study of representations of lie algebras. In the past few years the concept of wave front set d. Any nitedimensional real lie algebra is isomorphic to the lie algebra of a lie group. This gives a focused account of both the structure of compact lie groups and their irreducible necessarily finite dimensional representations, relative to the standard highest weight theory for a complex semisimple lie algebra. Before leaving the reals behind us, note the following theorems we will not prove. Complex geometry and representations of lie groups 205 subgroup b c g corresponding to a borel subalgebra b c g is defined to be the gnormalizer of b, that is, 1. Classical limit associated varieties deformation quantization howes wavefront. Wave front sets of reductive lie group representations ii american.

A tutorial on se3 transformation parameterizations and. In the archimedean case the wavefront set of a representation. Introduction let k be a global or nonarchimedean local. Realizing a lie group as an algebraic group is the. Representation theory of classical compact lie groups. We note that rand mhave 3 and 6 degrees of freedom respectively. Harmonically induced representations of solvable lie groups. Mixing, counting, and equidistribution in lie groups. Let cbe the field of complex numbers and fa vector space over c. Helgasons books differential geometry, lie groups, and symmetric spaces and groups and geometric analysis, intermixed with new content created for the class.

The object of the present paper is to make a deeper study of the representations of the algebra. Even before explaining the nature of the algorithm, it is worth recalling why this is an interesting question. The wave front set of an induced lie group representation. Representation theory university of california, berkeley. 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. As observed in gsa the depth can also be described in terms of the wavefront set of the representation. Introduction to representations theory of lie groups raul gomez october 14, 2009 introduction the purpose of this notes is twofold. Degenerate whittaker vectors and small automorphic. Wave front sets of representations of lie groups springerlink. Lie groups arise in a natural way as transformation groups of geometric objects. Semi groups and representations of lie groups 2 introduction thestudy of liesemi groups and their representations was initiated by e. For a large part, they summarise relevant material from knapps book 12. Representation theory of lie groups is the systematic study of symmetries and ways of exploiting them.

Most lie groups books fall into one of two categories. There arent very many geometrically flavored books on lie groups. 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. 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. Orbit method commuting algebras differential operator algebras hamiltonian gspaces 4. Wave front sets of reductive lie group representations project euclid. Unitaryrepresentationsofrealreductivegroups atlas of lie. I can think of only this one and compact lie groups by sepanski. Dec 30, 2014 raising nilpotent orbits in wavefront sets article pdf available in representation theory of the american mathematical society 2015 december 2014 with 20 reads how we measure reads. To answer this, we will show by examples what are the most important results of this theory, and the. Lie groups, lie algebras, and their representation graduate texts in mathematics, vol. Since the wave front set of a representation of a reductive group is a union of nilpotent coadjoint orbits in the dual g. Formally, an action of a group gon a set xis an action map a. The first contribution of this thesis is a new approach based on the theory of group representations in order to solve in a general an unified way the unitarization and inversion problems for.

Your story matters citation eskin, alex, and curtis t. For instance, they play a fundamental role in the seminal series of papers,, which give necessary and sufficient conditions for discrete decomposability of lie group representations. Lie groups and lie algebras for physicists harold steinacker lecture notes1, spring 2015 university of vienna fakult at fur physik universit at wien boltzmanngasse 5, a1090 wien, austria email. This is the third in a series of papers on wave front sets of lie group representations hho16, har. Wave front set, singular spectrum, analytic wave front set, reduc tive lie group, induced representation, tempered representation, branching problem. Efficient and robust registration on the 3d special. Any distribution on a manifold m has a welldefined wavefront set, which is. For a survey of the basic problems and results the reader is referred to that paper and to chapter xxv of 7. Raising nilpotent orbits in wavefront sets dihua jiang, baiying liu, and gordan savin abstract.

Lectures on lie groups and representations of locally compact. More formally, a representation means a homomorphism from the group to the automorphism group of an object. These notes are an introduction to lie algebras, algebraic groups, and lie groups in characteristic zero, emphasizing the relationships between these objects visible in their categories of representations. Introduction to representations theory of lie groups. Ramanan no part of this book may be reproduced in any form by print, micro. Representations of compact lie groups springerlink. Discrete means that the elements are not continuously connected.

Lie groups in physics1 version 250607 institute for theoretical physics utrecht university beta faculty 2007 english version by g. If we relax some commutativity assumptions, we have to replace group or lie algebra by hopf algebra. We study wavefront sets of representations of reductive groups over global or nonarchimedean local. The basic problem of representation theory is to classify all representations of a given group gup to isomorphisms. Wave front sets of reductive lie group representations iii. In fact, each complex lie algebra may be the complexi cation of several nonisomorphic real simple lie algebras. 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. Mixing, counting and equidistribution in lie groups alex eskin and curt mcmullen. The notes are selfcontained except for some details about topological groups for which we refer to chevalleys theory of lie. Lie groups, lie algebras, and their representation. First we deal with connected and simple connected lie groups in section 6, for which there is a representation of the lie group associated to each representation of the lie algebra. If g is a real, reductive algebraic group and \pi is a unitary representation of g that is weakly contained.

In mathematics and theoretical physics, a representation of a lie group is a linear action of a lie group on a vector space. 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. Representtionsa of lie algebras, with applications to arpticle physics james marrone university of chicago mathematics reu, august 2007 abstract. If youre perfectly happy with a more algebraic treatment, read no further. However, it does not suffice for the geometric realization of the discrete. Lie groups in physics1 institute for theoretical physics. It is possible to adopt a lie algebraic approach to the study of the general aspects of representation theory of lie groups.

Pdf file 1600 kb djvu file 388 kb article info and citation. Integral representations of these semi groups are given. The rst 4 groups are abelian, the last three are nonabelian. In this theory, one considers representations of the group algebra a cg of a. Representations of compact lie groups graduate texts in. Then gis generated by any open set containing the identity. Lie algebras, though mentioned occasionally, are not used in an essential way. Representation theory ct, lent 2005 1 what is representation theory.

It is an extremely important discipline in modern mathematics and has strong connections with and applications to such diverse fields as number theory and mathematical physics. It introduces the reader to the representation theory of compact lie groups. Lecture 1 representations of reductive lie groups notes lecture 1 128 this is math 224. In this paper it is shown that the wave front set of a direct integral of singular, irreducible representations of a real, reductive algebraic group is. For example, the symmetric group s n is the group of all permutations symmetries of 1. A great deal is known about such representations, a basic tool in. Wave front sets of representations of lie groups,automorphic forms, representation theory,and arithmetic, tata inst. Compact lie groups and representation theory lecture notes. Howes wavefront set v nice repn of nice lie group g.

Essays in the history of lie groups and algebraic groups. Wave front sets of reductive lie group representations. In the past few years the concept of wave front set d has proved fruitful for the theory of distributions and p. Remarks on the wavefront set of a distribution, trans. Introduction the study of lie semi groups and their representations was initiated by e. Wave front sets of reductive lie group representations benjamin harris, hongyu he, and gestur olafsson abstract. The ideas in this series are heavily influenced by the earlier work. We discuss the discrete groups that arise in this way in. Theory of lie groups was the first systematic exposition of the foundations of lie group theory consistently adopting the global viewpoint, based on the notion of analytic manifold. The character and the wave front set correspondence in the stable.

Anyone interested in understanding the connection between quarks and representation theory is an expeditious fashion should read this book. Every lie group has a lie algebra attached to it, and these algebras will also play an important role in the thesis. Group representations 6 completely reducible representations if a representation d. It seems it might also be of use in the representation theory of lie groups. A unipotent support for irreducible representations core. Eventually these notes will consist of three chapters, each about 100 pages long, and a short appendix. Princeton university, princeton nj 08544 university of california, berkeley, ca 94720 8 july, 1992 1 introduction. Representation theory of compact groups and complex reductive. Weyl group representations, nilpotent orbits, and the. The full set of symmetries of the square forms a group. The structure of lie groups and the classi cation of their representations are subjects. Separation of unitary representations of connected lie groups by their moment sets lobna abdelmoula, didier arnal, jean ludwig and mohamed selmi abstract.

G auth2 be a group of isometries of the hyperbolic plane h2 such that. Pdf wave front sets of reductive lie group representations. This means that the group space can be described by a set of coordinates. Eisenstein series and automorphic representations philipp fleig1. A third important issue we will be concerned with in this chapter is the relations of our general shearlet groups to other classical groups, in particular to the symplectic groups spd. Group representations 2 itfollowsthatthetranslationgroupadmitsanin. Depending on your own background, you might find the textbook by brocker and tom dieck useful. In such cases, the classification of representations reduces to the classification of irreducible representations.

Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. One can thus give an axiomatic characterization of the category of representations of a hopf algebra, and this is one of the most fruitful ways to deal with quantum groups. Compact lie groups and representation theory lecture notes lecturer. This book remained the basic reference on lie groups for at least two decades. Introduction to representation theory mit mathematics. This is one of the main reasons we will mostly be restricting our attention to representations of compact lie groups this semester. Wave front sets of reductive lie group representations article pdf available in duke mathematical journal 1655 august 20 with 61 reads how we measure reads. Representations of finite groups of lie type chapter 5 topological groups, representations, and haar measure. The wavefront set of a region d in r 2 is the set of pairs x. Groups arise in nature as sets of symmetries of an object, which are closed under composition and under taking inverses. Distribution on right above is smooth, so wavefront set is zero. Mixing, counting and equidistribution in lie groups. Certain types of lie groupsnotably, compact lie groupshave the property that every finitedimensional representation is isomorphic to a direct sum of irreducible representations. Tensor representations are given by the direct tensor product of copies of the vector representation.

The adelic methods are so powerful that one can obtain a closed, simple and group. In this work, we focus on the lie group lie algebra relationship and on the representation theory of lie groups through the representations of lie algebras. Merger 142 references for chapter vi 144 chapter vii. The rst goal is to give a quick answer to the question \what is representation theory about.

Lie group representations howes wavefront set and the size of representations associated varieties and the size of representations turning on your computer. 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. This is the third in a series of papers on wave front sets of lie group representations. The purpose here is to define and discuss general properties of wave front sets of representations, and to give some examples. Groups and symmetries is a short, concise book that provides an introduction to the subject of lie groups, lie algebras, their representations, and their uses in theoretical particle physics.

Equidistribution in lie groups the harvard community has made this article openly available. V admits an invariant subspace wv andmoreoveralsothecomplement w. Representations play an important role in the study of continuous symmetry. Se3 where so3 and se3 are the special orthogonal and the special euclidean groups respectively. This book has grown out of a set of lecture notes i had prepared for a course on lie groups in 1966. When i lectured again on the subject in 1972, i revised the notes substantially. Lecture notes introduction to lie groups mathematics. Among the relevant events related to this core topic we mention the lectures by karlheinz gr ochenig on \gabor analysis and its mysteries during the winter school \advances in mathematics of signal processing january 1115, 2016, the lecture series of. Fourier series finitediml representations abstract harmonic analysis quadratic forms 2. The size of infinitedimensional representations mit math. Thus, the set of all symmetry transformations that characterize the symmetry of a system, are elements of a group. Its close relative, the singular spectrum of a hyperfunction, has already been discussed in a special context in kv which served as the catalyst for this note. Lectures on lie groups and representations of locally compact groups by f.