G denote the group of maps from the unit circle s1 into g, of a suitable class. Representation theory of gl n over nonarchimedean local. Pdf computing the iwasawa decomposition of the classical lie. The subgroup n is normal in b and b 0, so these decompositions are semidirect products. Raghuram march 29, 2007 contents 1 introduction 1 2 generalities on representations 2 3 preliminaries on gl nf 6 4 parabolic induction 11 5 jacquet functors 15 6 supercuspidal representations 20 7 discrete series representations 26 8 langlands classi. Generalized dpw method and an application to isometric.
Cartan and iwasawa decompositions in lie theory 5 theorem 3. On convexity, the weyl group and the iwasawa decomposition. However, we can state that, every real semisimple lie algebra can be realized as a lie algebra of real matrices closed under transpose which will be clear after proposition 1. Iwasawa decomposition of a real semisimple lie algebra, the universal enveloping algebra of cj. Semisimple lie algebras and the root space decomposition ebrahim may 1, 2015 this document will develop just the material needed to describe a semisimple lie algebra in terms of its root space decomposition. In the iwasawa decomposition, neither knor annor aor n is normal in sl 2r. Semisimple lie algebras and the root space decomposition.
Pdf in this article, we show how the qr decomposition can be used to compute the iwasawa decomposition for all classical lie groups of. More specifically, let gnak be an iwasawa decomposition of a connected real semisimple lie group with finite center, and let m be the centralizer of a in k. Computing the iwasawa decomposition of the classical lie groups of noncompact type using the qr decomposition. Let wr denotethe projectionof umfi into ugdetermined by. Lectures on harmonic analysis on lie groups and related. Readings introduction to lie groups mathematics mit. The iwasawa decomposition kan kan for sl 2r is the analogue of the polar decomposition s1 r 0 for c. It is based on the canonical form of the lie algebra. Cartan decomposition on the lie algebra level 354 3. In fact, a choice of a topological generator g of g gives rise to an isomorphism z pjtk l. Uniqueness properties of the iwasawa decomposition 378 6. Cmjd, iwasawa decomposition, bruhat decomposition let g be a connected semisimple lie group having g as its lie algebra. The fact that k, a, nare banachlie groups with the corresponding lie algebras k, a, and n, respectively, and follows for instance by corollary 3.
Generalizations of the cartan and iwasawa decompositions for sl. The following result on the structure of lmodules allowed serre to rephrase the theorem of iwasawa. If time permits we will discuss some uniqueness properties of the iwasawa. We consider the computation of the iwasawa decomposition of a symplectic matrix via the qr factorization. The iwasawa type decomposition for states that for an open dense subset of one has 2, page 167. Given a real symplectic matrix s s11 s12 s21 s22, the following algorithm computes the factors k, a, n of the iwasawa decomposition of s. In the setting of a connected noncompact semisimple lie group with a finite center, we prove a property of the iwasawa decomposition, and use it to show that, in kostants convexity result, the iwasawa projection puts almost all its weight near a single vertex of the convex hull. Weight distribution of iwasawa projection sciencedirect. Iwasawa decomposition for any classical lie group of noncompact type using a device introduced in 9 for the group. Cartan involution and decomposition on lie algebra level mostly it will be assumed that g is a real lie algebra of matrices over r or c and closed under. Chapter vi structure theory of semisimple groups stony brook. The iwasawa decomposition of g is a group decomposition g nak, where nis a nilpotent analytic 1 subgroup, ais an abelian analytic subgroup, and kis a maximal compact analytic subgroup 7. The fact that k, a, nare banach lie groups with the corresponding lie algebras k, a, and n, respectively, and follows for instance by corollary 3.
Cmjd, iwasawa decomposition, bruhat decomposition let g be a connected noncompact semisimple lie group having g as its lie algebra. Let g be a noncompact real semisimple lie group with lie algebra g. Direct determination of the iwasawa decomposition for. A direct method for the determination of the iwasawa decomposition of any noncompact semisimple real lie algebra is described in detail.
Computing the iwasawa decomposition of the classical lie groups of. The cartan decomposition reduces the classification of real noncompact semisimple lie algebras to that of compact semisimple lie algebras and involutory automorphisms in them. G is the inverse limit of the group rings z p gh as h runs through the open normal subgroups of g. The iwasawa decomposition g kpis stronger yet than the existence of p. If g0 is a real semisimple lie algebra, then g0 has a cartan involution. The iwasawa decomposition theorem is a fundamental result in the theory of lie groups. We use, in particular, clifford algebra and the spin representation. G of a profinite group g is a variation of the group ring of g with padic coefficients that take the topology of g into account. Iwasawa factorization of the pseudoorthogonal group so 5, 1 and its corresponding lie algebra so 5, 1 springerlink. The foliations we wan t to describe arise from the iwasawa decomposition of a semisimple lie algebra the group level decomposition will also play a role. Iwasawa decompositions of some infinitedimensional lie groups 3 g kb.
Representation theory of gln over nonarchimedean local. As another application, we will revisit the results of huangtam, on the iteration of group elements under the. When g is a split real form, the foliations induced from the iwasawa decomposition are actually lagrangian brations with a global transverse lagrangian section. The regular coadjoint orbits of g a certain dense family of. Proposition every nontrivial simple lie algebra is semisimple. The cartan decomposition of a complex semisimple lie algebra. The physically important lie algebras so 3,1, so 4,1, so 3,2, and so 4,2 are treated as illustrative examples.
The iwasawa decomposition and the bruhat decomposition of the automorphism group on certain exceptional jordan algebra nishio, akihiro, tsukuba journal of mathematics, 20. In this article, we show how the qr decomposition can be used to compute the iwasawa decomposition for all classical lie groups of noncompact. Iwasawa decomposition of the lie group sl2,r worked out. Any two cartan involutions are conjugate via inn g0. Isotropic foliations of coadjoint orbits from the iwasawa decomposition william d. In mathematics, the iwasawa decomposition aka kan from its expression of a semisimple lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix a consequence of gramschmidt orthogonalization. Five lectures on lattices in semisimple lie groups 5 b1 a12a21 0. Lecture notes to the graduate course finite dimensional algebra during spring 2019 at. The cartan decomposition of a real reductive lie group g factors the group into hq where h is maximal compact and q is the symmetric space.
Let g k an be the iwasawa decomposition of a real lie group g with reductive lie algebra g. If g is the lie algebra of g then the cartan decomposition of g is given by g k. In this article, we show how the qr decomposition can be used to compute the iwasawa decomposition for all classical lie groups of noncompact type. G, is said to be of connection order ba if the fourier expansion in the loop parameter. We carry out a structural analysis of the lie group so5, 1 by obtaining its iwasawa factorization. The cartan decomposition generalizes the decomposition of a classical matrix lie algebra into its skewhermitian and hermitian parts. Pdf computing the iwasawa decomposition of the classical. In mathematics, the cartan decomposition is a decomposition of a semisimple lie group or lie algebra, which plays an important role in their structure theory and representation theory. If g is the lie algebra of g then the cartan decomposition of g is given by g. For instance, it is a key ingredient for results of furman, thm. Let k be a maximal combact subgroup of g and gkan be an iwasawa decomposition. A clifford algebra approach to real rank one simple lie.
In the present paper we study the iwasawa decomposition in the context of. Sophus lie and the role of lie groups in mathematics. Complexification of a simple real lie algebra 406 10. Universitetsforlaget scandinavian university press 1994.
Let gc be a simply connected lie group with lie algebra gc, where for a real lie algebra l,bylc we mean its complexi. Let gbe a noncompact real semisimple lie group with lie algebra g. This approach can also be used for the exceptional lie groups. On the iwasawa decomposition of a symplectic matrix. For example, the conjugate of an element of kby 1 1 0 1 is usually not in kand the conjugate of an element of an by 0 1 1 0 is usually not in an. Let g be the complexi cation of g0, and choose a compact real form u0 of g. Loops in noncompact groups of hermitian symmetric type and factorization caine, a and pickrell, d, journal of generalized lie theory and applications, 2015. Next theorem guarantees the existence of the socalled nonlinear iwasawa decomposition. Peter kellersch the iwasawa decomposition for the untwisted group of loops in semisimple lie groups.
In this article, we show how the qr decomposition can be. Notes to lie algebras and representation theory zhengyaowu abstract. Connection with the iwasawa and polar decompositions 28 chapter ii. The section computes appropriate haar measures and jacobians for the iwasawa decomposition. Find materials for this course in the pages linked along the left.
Computing the iwasawa decomposition of the classical lie. The decompositions you mention are wellknown in the theory of lie groups, e. The iwasawa decomposition holds for all semisimple lie groups with the same proof. A little bit more advanced video that deals with representation theory of lie groups, specifically the iwasawa decomposition of sl2,r, the special. Lie algebras and ados theorem princeton university. Let g be any of the classical groups appearing in column 1 of table. Kirwin abstract let g be a real semisimple lie group. Iwasawa decompositions will also be constructed at lie group and lie algebra level. R and let a be the subset of g consisting of diagonal matrices with positive diagonal entries, n the subset of g consisting of upper triangular matrices with 1 in each diagonal entry, and b the subset of g consisting of upper triangular matrices with.
Two mathematical sources worth consulting are helgason differential geometry, lie groups, and symmetric spaces chapter ix. Isotropic foliations of coadjoint orbits from the iwasawa decomposition. It is named after kenkichi iwasawa, the japanese mathematician who developed this method. Computing the iwasawa decomposition of the classical lie groups of noncompact type using the qr decomposition article pdf available in linear algebra and its applications 493. Cezar oniciuc tangency and harmonicity properties in romanian, ph. If both a12 and a21 are equal to 0, then the product. Journal of lie theory volume 12 2002 617618 c 2002 heldermann verlag remark on the complexi ed iwasawa decomposition andrew r. The action of a lie group g on a manifold m determines also vector. In this talk we will introduce the iwasawa decomposition of real semisimple lie groups. The harishchandra mapping to which we refer is the homomorphism p. In this section, we describe four wellknown decompositions of g or an element of g, namely the cartan decomposition, the iwasawa decomposition, the bruhat decomposition, and the complete multiplicative jordan decomposition. Browse other questions tagged linear algebra matrices functionalanalysis or ask your own question. An asymptotic result on the acomponent in the iwasawa.
The proof of the lie algebra decomposition just uses symmetric. K is the corresponding projection and \\bar n\ is the usual other n made with the opposite positivity then we show. It is, indeed, the special case of the iwasawa decomposition. The 1dimensional lie algebra is called the trivial simple lie algebra. The iwasawa factorization at the lie algebra level is also obtained. Stratonovich equation with the elements of the lie algebra acting on the left. We present practical algorithms to compute the semisimple and solvable parts starting from a basis of a given. Opening lecture by sigurdur helgason at a nordic teachers. Computing the iwasawa decomposition of the classical lie groups of noncompact type using the qr decomposition 2016, linear algebra and its applications download pdf view details. The question of knowing if the index were additive in the decomposition g0 k0.
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