Complex Adjoint Orbits in Lie Theory and Geometry
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Nilpotent Orbits and Weyl Group Representations, I
Nilpotent Orbits and Weyl Group Representations, I O.S.U. Lie Groups Seminar April 12, 2017 1. Introduction Today, G will denote a complex Lie group and g the complexification of its Lie algebra. Moreover, to keep matters simple, and in particular amenable to finite computations, we may as well restrict G to be the prototypical class of simple complex Lie groups. Then the relatively simple linear algebraic object g determines G up to isogeny (a choice of covering groups). In fact, an even simpler datum determines g up to isomorphism; that is its corresponding root system ∆; which in turn can be reduced to a even simpler combinatorial datum; a set Π of mutually obtuse linearly independent vectors in a real Euclidean space such that 2 hα; βi f2g if a = β hα_; βi ≡ 2 ; α; β 2 Π hα; αi f0; −1; −2; −3g if α 6= β Such simple systems are then reduced to simple weighted planar diagrams constructed by the following algorithm: • each vertex corresponds to a particular simple root α 2 Π • if hα_; βi= 6 0, then an edge is drawn between α and β. This edges are drawn between vertices whenever hα; βi 6= 0 according to the following rules { There is an undirected edge of weight 1 connecting α and β whenever hαv; βi = hβv; αi = −1 { There is a direced edge of weight 2 from α to β whenever hαv; βi = −1 and β_; α = −2 { There is a directed edge of weight 3 from α to β whenever hα_; βi = −1 and β_; α = −3 (The edge possibilities listed above are actually the only possibilities that occur when ha; βi 6= 0) If we demand further that for each α 2 Π, there is also a β 2 Π such that hα_; βi 2 {−1; −2; −3g (so that we consider only connected Dynkin diagrams), then one has only a set of five exceptional possibilities (G2;F4;E6;E7 and E8) and four infinite families (An;Bn;Cn;Dn) of simple systems. -
Coadjoint Orbits of Lie Algebras and Cartan Class
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 15 (2019), 002, 20 pages Coadjoint Orbits of Lie Algebras and Cartan Class Michel GOZE y and Elisabeth REMM z y Ramm Algebra Center, 4 rue de Cluny, F-68800 Rammersmatt, France E-mail: [email protected] z Universit´ede Haute-Alsace, IRIMAS EA 7499, D´epartement de Math´ematiques, F-68100 Mulhouse, France E-mail: [email protected] Received September 13, 2018, in final form December 31, 2018; Published online January 09, 2019 https://doi.org/10.3842/SIGMA.2019.002 Abstract. We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit O(α) at the point α corresponds to the characteristic space associated to the left invariant form α and its dimension is the even part of the Cartan class of α. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is 2 or 4. We determine also the Lie algebras of dimension 2n or 2n + 1 having an orbit of dimension 2n. Key words: Lie algebras; coadjoint representation; contact forms; Frobenius Lie algebras; Cartan class 2010 Mathematics Subject Classification: 17B20; 17B30; 53D10; 53D05 1 Introduction Let G be a connected Lie group, g its Lie algebra and g∗ the dual vector space of g. We identify g with the Lie algebra of left invariant vector fields on G and g∗ with the vector space of left invariant Pfaffian forms on G. -
Orbital Varieties and Unipotent Representations of Classical
Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups by Thomas Pietraho M.S., University of Chicago, 1996 B.A., University of Chicago, 1996 Submitted to the Department of Mathematics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2001 °c Thomas Pietraho, MMI. All rights reserved. The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part and to grant others the right to do so. Author ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Department of Mathematics April 25, 2001 Certified by :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: David A. Vogan Professor of Mathematics Thesis Supervisor Accepted by :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Tomasz Mrowka Chairman, Department Committee on Graduate Students 2 Orbital Varieties and Unipotent Representations of Classical Semisimple Lie Groups by Thomas Pietraho Submitted to the Department of Mathematics on April 25, 2001, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract Let G be a complex semi-simple and classical Lie group. The notion of a Lagrangian covering can be used to extend the method of polarizing a nilpotent coadjoint orbit to obtain a unitary representation of G. W. Graham and D. Vogan propose such a construction, relying on the notions of orbital varieties and admissible orbit data. The first part of the thesis seeks to understand the set of orbital varieties contained in a given nipotent orbit. Starting from N. Spaltenstein’s parameterization of the irreducible components of the variety of flags fixed by a unipotent, we produce a parameterization of the orbital varieties lying in the corresponding fiber of the Steinberg map. -
Nilpotent Orbits, Normality, and Hamiltonian Group Actions
JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 7, Number 2, April 1994 NILPOTENT ORBITS, NORMALITY, AND HAMILTONIAN GROUP ACTIONS RANEE BRYLINSKI AND BERTRAM KOSTANT The theory of coadjoint orbits of Lie groups is central to a number of areas in mathematics. A list of such areas would include (1) group representation theory, (2) symmetry-related Hamiltonian mechanics and attendant physical theories, (3) symplectic geometry, (4) moment maps, and (5) geometric quantization. From many points of view the most interesting cases arise when the group G in question is semisimple. For semisimple G the most familiar of the orbits are orbits of semisimple elements. In that case the associated representation theory is pretty much understood (the Borel-Weil-Bott Theorem and noncom- pact analogs, e.g., Zuckerman functors). The isotropy subgroups are reductive and the orbits are in one form or another related to flag manifolds and their natural generalizations. A totally different experience is encountered with nilpotent orbits of semi- simple groups. Here the associated representation theory (unipotent represen- tations) is poorly understood and there is a loss of reductivity of isotropy sub- groups. To make matters worse (or really more interesting) orbits are no longer closed and there can be a failure of normality for orbit closures. In addition simple connectivity is generally gone but more seriously there may exist no invariant polarizations. The interest in nilpotent orbits of semisimple Lie groups has increased sharply over the last two decades. This perhaps may be attributed to the recurring experience that sophisticated aspects of semisimple group theory often lead one to these orbits (e.g., the Springer correspondence with representations of the Weyl group). -
C*-Algebras and Kirillov's Coadjoint Orbit Method
C∗-ALGEBRAS AND KIRILLOV'S COADJOINT ORBIT METHOD DAVID SCHWEIN One of the main goals of representation theory is to understand the unitary dual of a topological group, that is, the set of irreducible unitary representations. Much of modern number theory, for instance, is concerned with describing the unitary duals of various reduc- tive groups over a local field or the adeles, and here our understanding of the representation theory is far from complete. For a different class of groups, the nilpotent Lie groups, A. A. Kirillov gave in the mid- nineteenth century [Kir62] a simple and transparent description of the unitary dual: it is the orbit space under the coadjoint action of the Lie group on the dual of its Lie algebra. The goal of this article, notes for a talk, is to explain Kirillov's result and illustrate it with the Heisenberg group, following Kirillov's excellent and approachable book on the subject [Kir04]. We begin with an introductory section on the unitary dual of a C∗-algebra, the proper setting (currently) for unitary representations of locally compact groups, following Dixmier's exhaustive monograph on C∗-algebras [Dix77]. 1. C∗-algebras and the unitary dual The theory of unitary representations of locally compact topological groups, for instance, reductive p-adic groups, is a special case of the more general theory of representations of C∗-algebras. In this section we review the representation theory of C∗-algebras and see how it specializes to that of topological groups. 1.1. Definitions and examples. A Banach algebra is a Banach space A equipped with an algebra structure with respect to which the norm is sub-multiplicative: ka · bk ≤ kak · kbk; a; b 2 A: We do note require Banach algebras to be unital, and in fact, we will see shortly that there are many natural examples that are not unital. -
Representation Theory
M392C NOTES: REPRESENTATION THEORY ARUN DEBRAY MAY 14, 2017 These notes were taken in UT Austin's M392C (Representation Theory) class in Spring 2017, taught by Sam Gunningham. I live-TEXed them using vim, so there may be typos; please send questions, comments, complaints, and corrections to [email protected]. Thanks to Kartik Chitturi, Adrian Clough, Tom Gannon, Nathan Guermond, Sam Gunningham, Jay Hathaway, and Surya Raghavendran for correcting a few errors. Contents 1. Lie groups and smooth actions: 1/18/172 2. Representation theory of compact groups: 1/20/174 3. Operations on representations: 1/23/176 4. Complete reducibility: 1/25/178 5. Some examples: 1/27/17 10 6. Matrix coefficients and characters: 1/30/17 12 7. The Peter-Weyl theorem: 2/1/17 13 8. Character tables: 2/3/17 15 9. The character theory of SU(2): 2/6/17 17 10. Representation theory of Lie groups: 2/8/17 19 11. Lie algebras: 2/10/17 20 12. The adjoint representations: 2/13/17 22 13. Representations of Lie algebras: 2/15/17 24 14. The representation theory of sl2(C): 2/17/17 25 15. Solvable and nilpotent Lie algebras: 2/20/17 27 16. Semisimple Lie algebras: 2/22/17 29 17. Invariant bilinear forms on Lie algebras: 2/24/17 31 18. Classical Lie groups and Lie algebras: 2/27/17 32 19. Roots and root spaces: 3/1/17 34 20. Properties of roots: 3/3/17 36 21. Root systems: 3/6/17 37 22. Dynkin diagrams: 3/8/17 39 23. -
The Solution to a Generalized Toda Lattice and Representation Theory
ADVANCES IN MATHEMATICS 34,195338 (1979) The Solution to a Generalized Toda Lattice and Representation Theory BERTRAM KOSTANT* Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 CONTENTS 0. Introduction 0.1. Statement of the main results 0.2. A brief description of Sections l-7 0.3. General comments 1. Poisson commutatierity of translated invariants 1.1. A formula for the Poisson bracket 1.2. The case of a subalgebra of a semi-simple Lie algebra 1.3. Relating cent 8’ to S(g)’ 1.4. The theorem [I’, Jf3 = 0 for Lie summands 1.5. The case of a Bore1 subalgebra where f = C e-,. 1.6. The commutativity of certain vector fields on a;’ 2. The variety Z of normalized Jacobi elements 2.1. The variety 8, = f + d and invariants I E So 2.2. The vector fields II, I E So, on Z 2.3. The subvarieties Z(y) Z; Z and IV-conjugacy 2.4. The Bruhat-Gelfand decomposition, the isomorphism j3,: Q* -+ Z(r), and the rationality of Z(y) in the complex case 2.5. The foliation Z = U Z(r) in the complex case 2.6. The isomorphism b(W): G1”,, + Z(y) in the complex case 3. The parametrization Z s H x A, in the real case 3.1. The polar decomposition and centralizers # 3.2. The relation Gv n RHN = G,u for y E Z and the isomorphism Z(y) z [w’ 3.3. The open Weyl chamber R, and Z 3.4. The normalizer G of 9 in Ad gc and the subset G(*) = s(K)G* C G 3.5. -
1. Introduction
RICHARDSON ORBITS FOR REAL CLASSICAL GROUPS PETER E. TRAPA Abstract. For classical real Lie groups, we compute the annihilators and asso ciated vari- eties of the derived functor mo dules cohomologically induced from the trivial representation. (Generalizing the standard terminology for complex groups, the nilp otent orbits that arise as such asso ciated varieties are called Richardson orbits.) We show that every complex sp ecial orbit has a real form which is Richardson. As a consequence of the annihilator calculations, we give many new in nite families of simple highest weight mo dules with ir- reducible asso ciated varieties. Finally we sketch the analogous computations for singular derived functor mo dules in the weakly fair range and, as an application, outline a metho d to detect nonnormality of complex nilp otent orbit closures. 1. Introduction Fix a complex reductive Lie group, and consider its adjoint action on its Lie algebra g. If q = l u is a parab olic subalgebra, then the G saturation of u admits a unique dense orbit, and the nilp otent orbits which arise in this way are called Richardson orbits (following their initial study in [R]). They are the simplest kind of induced orbits, and they play an imp ortant role in the representation theory of G. It is natural to extend this construction to the case of a linear real reductive Lie group G . Let g denote the Lie algebra of G , write g for its complexi cation, and G for the R R R complexi cation of G . Let denote the Cartan involution of G , write g = k p for R R the complexi ed Cartan decomp osition, and let K denote the corresp onding subgroup of G. -
Math.RT] 3 Feb 2006 .Tehiebr Group Heisenberg the 8
KYUNGPOOK Math J. 42(2002), 199-272 The Method of Orbits for Real Lie Groups Jae-Hyun Yang Department of Mathematics, Inha University, Incheon 402-751, Korea e-mail : [email protected] (2000 Mathematics Subject Classification: Primary 22-XX, 20C35.) In this paper, we outline a development of the theory of orbit method for representa- tions of real Lie groups. In particular, we study the orbit method for representations of the Heisenberg group and the Jacobi group. Contents 1. Introduction 2. Quantization 3. The Kirillov Correspondence 4. Auslander-Kostant’s Theorem 5. The Obstacle for the Orbit Method 5.1. Compact Lie Groups 5.2. Semisimple Lie Groups 6. Nilpotent Orbits and the Kostant-Sekiguchi Correspondence 6.1. Jordan Decomposition 6.2. Nilpotent Orbits 6.3. The Kostant-Sekiguchi Correspondence 6.4. The Quantization of the K-action (due to D. Vogan) 7. Minimal Representations (g,h) 8. The Heisenberg Group HR 8.1. Schr¨odinger Representations 8.2. The Coadjoint Orbits of Picture 9. The Jacobi Group J arXiv:math/0602056v1 [math.RT] 3 Feb 2006 9.1. The Jacobi Group G 9.1.1. The Standard Coordinates of the Jacobi Group GJ 9.1.2. The Iwasawa Decomposition of the Jacobi Group GJ 9.2. The Lie Algebra of the Jacobi Group GJ (Received: December 4, 2001. Revised: May 14, 2002.) Key words and phrases: quantization, the Kirillov correspondence, nilpotent orbits, the Kostant-Sekiguchi correspondence, minimal representations, Heisenberg groups, the Jacobi group. This work was supported by INHA UNIVERSITY Research Grant.(INHA-21382) 199 200 Jae-Hyun Yang 9.3. -
Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics
Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics William Donnelly,1 Laurent Freidel,1 Seyed Faroogh Moosavian,1;2 Antony J. Speranza1 1Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo ON, N2L 2Y5, Canada 2Department of Physics, McGill University, Ernest Rutherford Physics Building, 3600 Rue Univer- sity, Montr´eal, QC H3A 2T8, Canada E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite- dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner's famous classification of representations of the Poincar´egroup since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincar´egroup. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same arXiv:2012.10367v1 [hep-th] 18 Dec 2020 set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid. -
Nilpotent Orbits and Theta-Stable Parabolic Subalgebras
REPRESENTATION THEORY An Electronic Journal of the American Mathematical Society Volume 2, Pages 1{32 (February 3, 1998) S 1088-4165(98)00038-7 NILPOTENT ORBITS AND THETA-STABLE PARABOLIC SUBALGEBRAS ALFRED G. NOEL¨ Abstract. In this work, we present a new classification of nilpotent orbits in a real reductive Lie algebra g under the action of its adjoint group. Our classification generalizes the Bala-Carter classification of the nilpotent orbits of complex semisimple Lie algebras. Our theory takes full advantage of the work of Kostant and Rallis on p , the “complex symmetric space associated C with g”. The Kostant-Sekiguchi correspondence, a bijection between nilpotent orbits in g and nilpotent orbits in p , is also used. We identify a fundamental C set of noticed nilpotents in p and show that they allow us to recover all other C nilpotents. Finally, we study the behaviour of a principal orbit, that is an orbit of maximal dimension, under our classification. This is not done in the other classification schemes currently available in the literature. Introduction Let g be a semisimple Lie algebra and G its adjoint group. We say that an C C element x of g is nilpotent if and only if, ad : y [x; y] for all y g ,isa C x C nilpotent endomorphism of g . Kostant (see also Dynkin→ [Dy]) has shown,∈ in his C fundamental 1959 paper [Ko], that the number of nilpotent orbits of G in g is C C finite. The Bala-Carter classification can be expressed as follows: There is a one-to-one correspondence between nilpotent orbits of g and con- C jugacy classes of pairs (m; p ), where m is a Levi subalgebra of g and p is a m C m distinguished parabolic subalgebra of the semisimple algebra [m; m]. -
Arxiv:1507.06234V4 [Math.RT] 2 Oct 2017 G a Endn Netnigti Motn Eutt H Modul the to Result Important This Extending on Done Been Has Oeciia Otiuin.I [ in Contributions
THE JACOBSON–MOROZOV THEOREM AND COMPLETE REDUCIBILITY OF LIE SUBALGEBRAS DAVID I. STEWART AND ADAM R. THOMAS* Abstract. In this paper we determine the precise extent to which the classical sl2-theory of complex semisimple finite-dimensional Lie algebras due to Jacobson–Morozov and Kostant can be extended to positive characteristic. This builds on work of Pommerening and improves significantly upon previous attempts due to Springer–Steinberg and Carter/Spaltenstein. Our main advance arises by investigating quite fully the extent to which subalgebras of the Lie algebras of semisimple algebraic groups over algebraically closed fields k are G-completely reducible, a notion essentially due to Serre. For example, if G is exceptional and char k = p ≥ 5, we classify the triples (h, g,p) such that there exists a non-G-completely reducible subalgebra of g = Lie(G) isomorphic to h. We do this also under the restriction that h be a p-subalgebra of g. We find that the notion of subalgebras being G-completely reducible effectively characterises when it is possible to find bijections between the conjugacy classes of sl2-subalgebras and nilpotent orbits and it is this which allows us to prove our main theorems. For absolute completeness, we also show that there is essentially only one occasion in which a nilpotent element cannot be extended to an sl2-triple when p ≥ 3: this happens for the exceptional orbit in G2 when p = 3. 1. Introduction The Jacobson–Morozov theorem is a fundamental result in the theory of complex semisimple Lie algebras, due originally to Morozov, but with a corrected proof by Jacobson.