The fundamental result is formulated in the introduction. In this note we deal with minimal unit vector elds on g. It is called reductive if its lie algebra is a direct sum of simple and trivial onedimensional lie algebras. In this paper a compact and semisimple lie group g is considered endowed with a 2plectic structure. Five lectures on lattices in semisimple lie groups 5 b1 a12a21 0. Algebraic dmodules and representation theory of semisimple. If gis compact, then it has a complexi cation g c, which is a complex semisimple lie group, and the irreducible unitary representations of gwhich turn out to always be nitedimensional lie. A representation a is of degree 1 with infinite image, if and only if ax is a circle group. There exists an invariant under right and left multiplication measurer dg on g. Coadjoint orbits of semisimple lie groups we start with recalling the notion of a coadjoint orbit. Ergodic actions of semisimple lie groups on compact principal.
Lectures on lie groups and representations of locally compact. If both a12 and a21 are equal to 0, then the product. Pdf on compact semisimple lie groups as 2plectic manifolds. We construct an assembly map in the framework of kktheory. A connected compact complex lie group is precisely a complex torus not to be confused with the complex lie group. The global classification of non compact real semisimple lie groups can be carried out in a similar but more complicated way. For example, su n and o n are semisimple, but u n is not, since the scalar matrices in u n form an abelian ideal.
More generally, we define a lie group to be semisimple if its lie algebra is semisimple and it has a faithful finitedimensional complex representation. Representation theory of compact groups and complex. On the fourier transform of a compact semisimple lie group volume 56 issue 1 n. The shape of f and the lattice symmetry of fm is determined by a compact semisimple lie group of rank n underlying the construction. Pdf a generalization of maximal functions on compact. In this paper, we consider a natural subelliptic structure in semisimple, compact and connected lie groups, and estimate the constant in the socalled. On representations of complex semisimple lie groups school of. The density of fm is fixed by our choice of a positive integer m, where 1. Here, semisimple means, that the corresponding lie algebra can be written as a direct sum of simple lie. Our first theorem shows that the lie algebra of a compact lie group. M semisimple compact lie groups we return now to considering a general.
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. This is employed to yield a variety of new examples from the compact lie groups son, sun and spn equipped with their standard. Translationinvariant function algebras 1097 let x be a compact connected group. Deformation quantization in the teaching of lie group representations balsomo, alexander j. If the killing form of a lie algebra is negative definite, then the lie algebra is the lie algebra of a compact semisimple lie group. Let pbe an odd very good prime for g, and let g c be a. Ergodic actions of semisimple lie groups on compact. In chapter 2 we describe the equivariant ktheory of the real semisimple lie group which acts on the complex ag arietvy of its complexi cation group.
Readings introduction to lie groups mathematics mit. There it is stated as a fact without proof, that any compact semisimple lie group has finite center. Pdf subelliptic estimates on compact semisimple lie groups. Any advance knowledge of complex semisimple lie algebras, universal enveloping algebras, and representation theory of finite or compact groups would. However in the case of compact lie groups, answers to most of these ques. Nov 20, 2008 an efficient construction of lattice points fm of any density and any admissible symmetry is described in a finite region f. Description and statement of the results let x be a riemannian manifold of dimension d and let g be a compact semisimple lie group with lie algebra g. I am reading introductory lecture notes on lie groups and lie algebras. X is an inverse system of compact lie groups with x as inverse limit, and.
Let k be a connected compact semisimple lie group, g its complexi. 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. A new algorithm is described for splitting class functions of an arbitrary semisimple compact lie group k into sums of irreducible characters. Subelliptic estimates on compact semisimple lie groups article pdf available in nonlinear analysis 7414. In addition, when gis a nonsimplylaced group, we show that there are three. We show that the lie group of 2plectomorphisms of g is finite.
Kramer advances in mathematics 228 2011 26232633 if s is a compact simple lie group, then every open subgroup g of autlies is compact by the remarks in notation 1. On the representation theory of semisimple lie groups. On other lie groups with left invariant metrics, they have been studied in 3, 6, 9. Finally we classify the closed translationinvariant function algebras on compact simple simply connected lie groups, as an example of a general enumeration procedure. Let gbe a compact connected semisimple lie group endowed with a biinvariant riemannian metric. During the past few years several authors have studied unitary representations of a lie group or, more generally, of a locally compact group on a hubert space. In the present paper we approach this problem from a somewhat different point of view. A complex semisimple lie group is a linear algebraic group. T eg be the lie algebra of g and let s t1 e g denote the unit sphere in g.
The topology of a semisimple lie group is essentially unique. In mathematics, a lie algebra is semisimple if it is a direct sum of simple lie algebras, i. Therefore we have the following consequence of theorem 7. Representations of semisimple lie groups u of u math. Semisimple compact lie groups we return now to considering a general. Subelliptic estimates on compact semisimple lie groups. A compact lie group is semisimple if its lie algebra is semisimple. Geometry and topology of coadjoint orbits of semisimple lie groups bernatska, julia and holod, petro, 2008. Computation of character decompositions of class functions on. For any compact lie group k, there is a smooth, irreducible, volumepreserving action of g on some principal kbundle p over some manifold m. Invariant einstein metrics on certain compact semisimple lie.
Structure theory of semisimple lie groups stony brook mathematics. 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. Let gbe a compact padic analytic group whose lie algebra is split semisimple. The representation theory of complex semisimple lie groups has a long history beginning in 1950 with the fundamental work of gelfand and. Computation of character decompositions of class functions on compact semisimple lie groups by r. Let h denote the algebraic maximal torus of g with lie algebra h. Qforms of real representations of compact semisimple lie groups after raghunathan and eberlein dave witte department of mathematics oklahoma state university stillwater, ok 74078 notation. The prove relies on a careful study of the orbits of the real group action on. In the algebra there are many abelian subalgebras, though not invariant. Computation of character decompositions of class functions. 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.
On the fourier transform of a compact semisimple lie group. Introduction let g0 be a connected semisimple lie group with. We establish contractions of the unitary irreducible representations of a compact semisimple lie group to the unitary irreducible representations of a heisenberg group by means of berezin quantization. A generalization of maximal functions on compact semisimple. The first point is with the definition that a compact lie algebra is the lie algebra of a compact lie group. Here, semisimple means, that the corresponding lie algebra can be written as a direct sum of simple lie algebras having no nontrivial ideals. Then we generalize this result about compact semisimple lie algebras to the class of nonhermitian real semisimple lie algebras having full rank. The structure of these module categories is the most intricate when the lie algebra of the group gis semisimple, so we focus on this case. The existence of this measure called the haar measure allows us to prove that the category of representations of gis semisimple. Introduction the following fundamental theorem was formulated in 1 let g be a semisimple algebraic real lie group of rank greater than 1, let. The goal is to cover the structure theory of semisimple compact connected lie groups, an.
On the volume of unit vector fields on a compact semisimple. For such a lie algebra, if nonzero, the following conditions are equivalent. Lie groups richard borcherds, mark haiman, nicolai reshetikhin, vera serganova, and theo johnsonfreyd october 5, 2016. Sharp fourier type and cotype of lebesgue spaces and schatten classes with respect to an arbitrary compact semisimple lie group are investigated. This is possible because the complexification of a real semisimple lie algebra is also semisimple see 1. Reductive groups occur naturally as symmetries of a number of mathematical objects in algebra. However, the correspondence between real and complex semisimple lie algebras established with the help of the complexification is not onetoone.
Let g t eg be the lie algebra of g and let s t1 e g denote the unit sphere in g. Let t be the maximal torus of g, and h be the maximal commutative subalgebra also called a cartan subalgebra of g. Any finite group may be given the structure of a complex lie group. On irreducible representations of compact padic analytic. The third point is just an application of the cauchyschwarz inequality. In general, the lie algebra of a compact lie group decomposes as the lie algebra direct sum of a commutative summand for which the corresponding subgroup is a torus and a summand on which the killing form is. Is every connected complex semisimple lie group the. Invariant einstein metrics on certain compact semisimple. Representation theory of compact groups and complex reductive. The four sections treat complex semisimple lie algebras, finitedimensional representations of complex semisimple lie algebras, compact lie groups and real. Let gbe a compact semisimple classical lie group, g denote the corresponding lie algebra, and g denote the dual space of g. The ultimate goal is to be able to understand all the irreducible unitary representations of any such group gup to unitary equivalence. For example, any one dimensional subspace is an abelian subalgebra.
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