DOCSLIB.ORG

ON the POSITIVITY of KIRILLOV's CHARACTER FORMULA Kirillov's

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
ON the POSITIVITY of KIRILLOV's CHARACTER FORMULA Kirillov's ON THE POSITIVITY OF KIRILLOV'S CHARACTER FORMULA EHSSAN KHANMOHAMMADI Abstract. We give a direct proof for the positivity of Kirillov's character on the convolution algebra of smooth, compactly supported functions on a connected, simply connected nilpotent Lie group G. Then we use this positivity result to construct a representation of G×G and establish a G×G-equivariant isometric isomorphism between our representation and the Hilbert{Schmidt operators on the underlying representation of G. Moreover, we show that the same methods apply to coadjoint orbits of other groups such as SL(2; R) under additional hypotheses that are automatically satisfied in the nilpotent case. Kirillov's character formula, Coadjoint orbit, Nilpotent Lie group, Positivity, Quantization, Polarization, The GNS construction. AMS MSC: 22E30, 22E27, 46L05. 1 1. Introduction 2 In his fundamental paper [12], Kirillov proved that coadjoint orbits of a con- 3 nected, simply connected nilpotent Lie group correspond, under quantization, to 4 the equivalence classes of its irreducible unitary representations. The theory of 5 geometric quantization due to Kirillov, Kostant [16], and Souriau [22] has shown 6 that this close connection extends to many other groups. 7 Kirillov's character formula says that the characters of irreducible unitary rep- 8 resentations of a Lie group G \should" be given by an equation of the form Z −1=2 i`(X)+σ (1.1) Tr πO(exp X) = j (X) e O ∗ 9 where O is the coadjoint orbit in g corresponding to πO 2 Gb, σ is a canonical 10 symplectic measure on O, and j is the analytic function on g defined by the formula sinh(ad X=2) j(X) = det : ad X=2 11 Kirillov proved his character formula for simply connected nilpotent and simply 12 connected compact Lie groups [12, 13] and conjectured its universality. The valid- 13 ity of this conjecture has been verified for some other classes of Lie groups, most 14 notably for the case of tempered representations of reductive Lie groups by Ross- 15 mann [20]. Moreover, Atiyah{Bott [2] and Berline{Vergne [3] following the work 16 of Duistermaat{Heckman [6], have shown that for compact Lie groups, Kirillov's 17 character formula is equivalent to the Weyl character formula. 18 Our main contributions in this paper are as follows. Rather than construct 19 representations, compute their characters, and compare the results with Kirillov's 20 character formula, we work directly with Kirillov's formula and give direct argu- 21 ments to prove positivity in the sense of operator algebras. More precisely, we give 1 2 EHSSAN KHANMOHAMMADI 22 a direct proof for the positivity of Kirillov's character on the convolution algebra 23 of the Schwartz functions on a connected nilpotent Lie group (Theorem 3.8). The 24 fact that Kirillov's character defines a positive trace is remarkable in light of the 25 Gelfand{Naimark{Segal (GNS) construction from operator algebra theory, since 26 GNS implies, roughly speaking, that any positive linear functional on a Lie group 27 is the character of a group representation. In the last section, we use our positivity 28 result and the GNS to construct a representation of G × G for a connected, simply 29 connected nilpotent Lie group G. Then in Theorem 5.3 we discuss the relation be- 30 tween the representations that we have obtained from the GNS construction, and 31 the Kirillov representations corresponding to the coadjoint orbits of G. Indeed, we 32 prove that there is a G × G-equivariant isometric isomorphism between our repre- 33 sentation and the Hilbert{Schmidt operators on the underlying representation of G. 34 We deal with nilpotent groups this way, thereby recovering some of the first results 35 obtained by Kirillov. As seen in Theorem 4.9, the same methods apply to coadjoint 36 orbits of other groups such as SL(2; R) in the presence of supplementary geomet- 37 ric hypotheses on the orbits that are automatically satisfied in the nilpotent case. 38 One hypothesis is the existence of a real polarization; a second is a completeness 39 hypothesis on the (locally affine) fibers of the associated Lagrangian fibration. 40 2. Some Notations and Background Material 41 In this section we fix some notations and collect several standard results that we 42 will use freely in the sequel. 43 Definition 2.1. Let G be a locally compact group G with a left Haar measure dµ. 44 The modular function ∆: G ! (0; 1) is defined via Z Z ∆(y) f(xy) dµ(x) = f(x) dµ(x) G G 1 45 for x and y in G and all f 2 L (G) so that Z Z f(x−1) dµ(x) = f(x)∆(x−1) dµ(x): G G −1 46 If G is a Lie group, then it is well known that ∆(g) = j det Ad(g )j. This 47 implies, for instance, that any connected nilpotent Lie group is unimodular, that 48 is, for such groups the modular function is constant and everywhere equal to one. Definition 2.2. Let G be a locally compact group with a left Haar measure dµ. Given functions f and g on G, their convolution f ∗ g is the function on G defined by Z f ∗ g(x) = f(y)g(y−1x) dµ(y) G Z = f(xy)g(y−1) dµ(y) G 49 whenever one (and hence both) of these integrals makes sense. 50 Let (π; Vπ) be a unitary representation of a locally compact group G. Then π 1 51 induces a continuous homomorphism of Banach ∗-algebras from L (G) to the space ON THE POSITIVITY OF KIRILLOV'S CHARACTER FORMULA 3 52 of bounded operators B(Vπ) via Bochner integration. By abuse of notation, we 53 denote this induced ∗-homomorphism by π. Hence, Z 1 π : L (G) !B(Vπ) and π(f) = f(g)π(g) dg: G 1 54 Here we are using the facts that L (G) is a Banach algebra under convolution, 1 55 which is just the usual group ring CG if G is finite, and that L (G) has a natural ∗ 56 isometric (conjugate-linear) involution f 7! f , where f ∗(x) = f(x−1)∆(x−1): 57 Note that these definitions are formulated so that f ∗g = L(f)(g) for the left regular 1 58 representation (L; L (G)). 59 Definition 2.3. We say that an irreducible unitary representation π of a Lie group 60 G has a global or Harish-Chandra character Θ if π(f) is of trace class for all f 2 1 61 Cc (G) and moreover f 7! Θ(f) = Tr π(f) is a distribution. 62 The next two results provide sufficient conditions for π(f) to be trace class. 63 Theorem 2.4 (Harish-Chandra [8]). Let G be a connected semisimple Lie group 64 with finite center. Then for every irreducible unitary representation π of G and 1 65 every f 2 Cc (G), the operator π(f) is trace class and the map f 7! Tr π(f) is a 66 distribution on G. 67 Theorem 2.5 (Kirillov [12]). Let G be a nilpotent Lie group. Then for every 68 irreducible unitary representation π of G and every Schwartz function f 2 S(G), 69 the operator f 7! Tr π(f) is trace class and the map f 7! Tr π(f) is a tempered 70 distribution on G. 71 Let (π; Vπ) be an irreducible unitary representation. Observe that the above ∗ ∗ 72 discussions imply that π(f ∗ f ) = π(f)π(f) . Thus, the distributional character 73 of (π; Vπ), whenever defined, is convolution-positive (or positive for short) in the 74 sense that ∗ ∗ 2 (2.1) Tr π(f ∗ f ) = Tr π(f)π(f) = kπ(f)kHS ≥ 0: 75 To close this section, we introduce Kirillov's character formula which says that the 76 characters χ of irreducible unitary representations of a Lie group G \should" be 77 given by an equation of the form Z −1=2 i`(X) (2.2) χ(exp X) = j (X) e dµO(`): O ∗ 78 Here O is a coadjoint orbit in g , µO is the canonical (or Liouville) symplectic 79 measure on O, and j is the Jacobian of the exponential map exp: g ! G given by 80 the formula sinh(ad X=2) j(X) = det ad X=2 81 whenever G is unimodular. This character formula should be interpreted as an 82 equation of distributions on a certain space of test functions on g as follows. For 83 all smooth functions f compactly supported in a sufficiently small neighborhood of 84 the origin in g, Z Z Z i`(X) −1=2 (2.3) Tr f(X)π(exp X) dX = e f(X)j (X) dXdµO(`) g O g 4 EHSSAN KHANMOHAMMADI 85 where π is the representation with character χ. 86 Theorem 2.6 (Kirillov [12]). Suppose G is a connected nilpotent Lie group. Then 87 the irreducible unitary representations of G are in natural one-to-one correspon- 88 dence with the integral orbits of G. Moreover, if G is simply connected and π is an 89 irreducible unitary representation of G with the associated coadjoint orbit O, then 90 (2.3) holds. 91 3. Nilpotent Lie Groups 92 Nilpotent Lie groups and their representations have been studied extensively in 93 the literature. From the many papers by Corwin, Greanleaf, Lipsman, Pukanzsky 94 and others we only cite [15], [18], and [21] and refer the reader to [4] for a more 95 comprehensive list of bibliographies.
Load more

Recommended publications
  • 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.
    [Show full text]
  • Lectures on the Orbit Method, by AA
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 42, Number 4, Pages 535–544 S 0273-0979(05)01065-7 Article electronically published on April 6, 2005 Lectures on the orbit method, by A. A. Kirillov, Graduate Studies in Mathemat- ics, vol. 64, American Mathematical Society, Providence, RI, 2004, xx+408 pp., $65.00, ISBN 0-8218-3530-0 1. Introduction without formulas This book is about a wonderfully successful example of (if you will forgive some geometric language) circular reasoning. Here is the short version. Everywhere in mathematics, we find geometric objects M (like manifolds) that are too complicated for us to understand. One way to make progress is to introduce a vector space V of functions on M.ThespaceV may be infinite-dimensional, but linear algebra is such a powerful tool that we can still say more about the function space V than about the original geometric space M. (With a liberal interpretation of “vector space of functions”, one can include things like the de Rham cohomology of M in this class of ideas.) Often M comes equipped with a group G of symmetries, but G and M may be even less comprehensible together than separately. Nevertheless, G will act on our function space V by change of variables (giving linear transformations), and so we get a representation of G on V . Our original (and impossible) problem of understanding all actions of G on geometric spaces M is therefore at least related to the problem of understanding all representations of G. Because this is a problem about vector spaces, it sounds a bit less daunting.
    [Show full text]
  • Sdiff(S ) and the Orbit Method Arxiv:1806.05235V2 [Hep-Th]
    SDiff(S2) and the orbit method Robert Penna Institute for Advanced Study, Princeton, NJ 08540, USA E-mail: [email protected] Abstract: The group of area preserving diffeomorphisms of the two sphere, SDiff(S2), is one of the simplest examples of an infinite dimensional Lie group. It plays a key role in incompressible hydrodynamics and it recently appeared in general relativity as a subgroup of two closely related, newly defined symmetry groups. We investi- gate its representation theory using the method of coadjoint orbits. We describe the Casimir functions and the Cartan algebra. Then we evaluate the trace of a simple SDiff(S2) operator using the Atiyah-Bott fixed point formula. The trace is divergent but we show that it has well-defined truncations related to the structure of SDiff(S2). Finally, we relate our results back to the recent appearances of SDiff(S2) in black hole physics. arXiv:1806.05235v2 [hep-th] 27 Nov 2019 Contents 1 Introduction1 2 Labeling the states2 2.1 Coadjoint orbits and Casimir functions3 2.2 SO(3) revisited5 2.3 Cartan algebra6 3 Traces8 3.1 SDiff(S2) 10 3.2 Fixed points 11 3.3 Complex coordinates 11 3.4 Result and discussion 12 4 Discussion 13 1 Introduction The set of area preserving diffeomorphisms of the two-sphere, SDiff(S2), is a basic example of an infinite dimensional Lie group. It is the configuration space and the symmetry group of the incompressible Euler equations for fluid flow on the sphere [1–5]. It may have a role to play in black hole physics, where it is a subgroup of two recently introduced symmetry groups.
    [Show full text]
  • Quantization, the Orbit Method, and Unitary Representations
    Quantization, the orbit method, and unitary representations David Vogan Quantization, the orbit method, and Physics Representations unitary representations Orbit method Hyperbolic orbits Elliptic orbits David Vogan Department of Mathematics Massachusetts Institute of Technology Representation Theory, Geometry, and Quantization: May 28–June 1 2018 Quantization, the Outline orbit method, and unitary representations David Vogan Physics: a view from a neighboring galaxy Physics Representations Orbit method Classical representation theory Hyperbolic orbits Elliptic orbits History of the orbit method in two slides Hyperbolic coadjoint orbits for reductive groups Elliptic coadjoint orbits for reductive groups Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time evolution ! unitary group t 7! exp(tA0) Observable A conserved ! [A0; A] = 0 Moral of the story: quantum mechanics is about Hilbert spaces and Lie algebras. Quantization, the Quantum mechanics orbit method, and unitary representations David Vogan ! H Physical system complex Hilbert space Physics States ! lines in H Representations Orbit method Observables ! linear operators fAj g on H Hyperbolic orbits Elliptic orbits Expected value of obs A !h Av; vi Energy ! special skew-adjoint operator A0 Time
    [Show full text]
  • Notes on the Orbit Method and Quantization
    Notes on the Orbit Metho d and Quantization Reyer Sjamaar Les Diablerets March 1 Contents Orbit metho d and quantization Index theorem in symplectic geometry 2 Orbit metho d and quantization A A Kirillov G Lie group innitedimensional group quan tum group Category of unitary representations of G Objects continuous homomorphisms T G UH H a Hilb ert space Morphism intertwining op erator from T to T continuous linear A H H A H H T g T g 2 1 A H H 3 Example X Gmanifold with Ginvariant mea sure Unitary representation on L X T g f x f g x Map F X X induces intertwining map F L X L X if is absolutely continuous wrt F T is indecomp osable if T T T for nonzero T and T T is irreducible if do es not have nontrivial invariant subspaces For unitary representation irreducible in decomp osable Unirrep unitary irreducible representation 4 Main problems of representation theory Describ e unitary dual G funirreps of Ggequivalence Decomp ose any T into unirreps Z T g T g dy y Y Sp ecial cases for H G closed littl e group G T a for T G decomp ose restriction Res H G S b for S H decomp ose induction Ind H Compute character of T G 5 Ad b let S H UH Supp ose GH has G G invariant measure Ind S L sections of H H G H Obtained by taking space of functions f G H satisfying f g h S hf g and completing wrt inner product Z hf f i hf x f xi dx H GH Ad let C G Put Z T g T g dg G With luck T H H is of trace class and Tr T is a distribution on G the charac
    [Show full text]
  • Interaction Between Lie Theory and Algebraic Geometry
    INTERACTIONS BETWEEN LIE THEORY AND ALGEBRAIC GEOMETRY SHILIN YU 1. Introduction My primary research interests lie in the interactions of complex/algebraic geometry with Lie the- ory and representation theory in the spirit of noncommutative geometry, derived algebraic geometry and mathematical physics. Both Lie theory and algebraic geometry have been at the center of the 20th-century mathematical studies. Entering the 21th century, the two subjects came across each other in a surprising way, due to the phenomenal work of Kontsevich on deformation quantization of Poisson manifolds ([45]) and the birth of derived algebraic geometry ([46]). We start from the following observation: in algebraic geometry, the well-known Grothendieck- Riemann-Roch (GRR) theorem involves the Todd class, which is a characteristic class defined by a certain formal power series. The same type of class also appears in the Atiyah-Singer index theorem, which is a generalization of the Riemann-Roch theorem and one of the most important theorems of the 20th century. On the other hand, the exact same power series also shows up in Lie theory in a fundamental way, long before the GRR theorem was formulated. Namely, it appears in in the differential of the exponential map of a Lie group the Baker-Campbell-Hausdorff (BCH) formula in Lie theory. It (or its inverse) also plays an essential role in the Duflo-Kirillov isomorphism ([28]). Due to the work of many people, including but not limited to Kapranov [40], Kontsevich [45], Markarian [48], and Ramadoss [53], it is now clear that the same power series appearing in the two seemly unrelated cases is not a coincidence.
    [Show full text]
  • Review of “Lectures on the Orbit Method,” by Aa Kirillov
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 00, Number 0, 1997 REVIEW OF \LECTURES ON THE ORBIT METHOD," BY A. A. KIRILLOV DAVID A. VOGAN, JR. Contents 1. Introduction without formulas 2. Introduction with formulas 3. Which orbits count? 4. Kirillov's book 1. Introduction without formulas This book is about a wonderfully successful example of (if you will forgive some geometric language) circular reasoning. Here is the short version. Everywhere in mathematics, we find geometric objects M (like manifolds) that are too complicated for us to understand. One way to make progress is to introduce a vector space V of functions on M. The space V may be infinite-dimensional, but linear algebra is such a powerful tool that we can still say more about the function space V than about the original geometric space M. (With a liberal interpretation of \vector space of functions," one can include things like the de Rham cohomology of M in this class of ideas.) Often M comes equipped with a group G of symmetries, but G and M may be even less comprehensible together than separately. Nevertheless, G will act on our function space V by change of variables (giving linear transformations), and so we get a representation of G on V . Our original (and impossible) problem of understanding all actions of G on geometric spaces M is therefore at least related to the problem of understanding all representations of G. Because this is a problem about vector spaces, it sounds a bit less daunting. The goal of the orbit method is to say something about all representations of a Lie group G.
    [Show full text]
  • P-Mechanics As a Physical Theory: an Introduction
    This is a repository copy of p-mechanics as a physical theory: an introduction. White Rose Research Online URL for this paper: http://eprints.whiterose.ac.uk/111528/ Version: Accepted Version Article: Kisil, VV orcid.org/0000-0002-6593-6147 (2004) p-mechanics as a physical theory: an introduction. Journal of Physics A: Mathematical and General, 37 (1). pp. 183-204. ISSN 0305-4470 https://doi.org/10.1088/0305-4470/37/1/013 © 2004 IOP Publishing Ltd. This is an author produced version of an article published in Journal of Physics A: Mathematical and General. Uploaded in accordance with the publisher's self-archiving policy. Reuse Items deposited in White Rose Research Online are protected by copyright, with all rights reserved unless indicated otherwise. They may be downloaded and/or printed for private study, or other acts as permitted by national copyright laws. The publisher or other rights holders may allow further reproduction and re-use of the full text version. This is indicated by the licence information on the White Rose Research Online record for the item. Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing [email protected] including the URL of the record and the reason for the withdrawal request. [email protected] https://eprints.whiterose.ac.uk/ LEEDS-PURE-MATH-200 2 -31, 200 2 arXiv:quant-ph/0212101 p-MECHANICS AS A PHYSICAL THEORY. AN INTRODUCTION VLADIMIR V. KISIL Abstract. The paper provides an introduction into p-mechanics, which is a consistent physical theory suitable for a simultaneous description of classical and quantum mechanics.
    [Show full text]
  • The Orbit Method and Its Applications Vignon Oussa Dalhousie
    The orbit method and its applications Vignon Oussa Dalhousie Mathematics Colloquium 1 Content This presentation is brief survey of Lie theory with an emphasis on nilpotent Lie groups, and how the orbit methods is exploited for the pur- pose of Harmonic analysis on nilpotent Lie groups. If you are an expert in Lie theory, half of the pre- sentation will be trivial. The other half which fo- cuses on the orbit method will probably be new to you if you are not in representation theory. If you are not an expert in both Lie theory and rep- resentation theory, my hope is that this presenta- tion will show you the beauty that we see in these theories, and convince you to join the rest of us in the adventure. 2 Acknowledgment I would like to thank Professor Keith Taylor who was very generous with his time, and for host- ing me during my visit at Dalhousie University. I have had a wonderful time. I also thank, Profes- sor Mahya Ghandehari and Mr. Joshua MacArthur for the wonderful discussions, and for welcoming during the 3 weeks that I have spent here in Halifax, NS, Canada. 3 A starting point A real Lie group is a set with two structures. First G is a group and G is manifold. These struc- tures agree in the sense that multiplication and inversion maps are smooth. That is 1. (x, y) 7! xy is a smooth map 2. x 7! x−1 is a smooth map 4 Some notations The general Linear group over the real num- bers denoted GL (n, R) is the group of all n × n invertible matrices with real entries.
    [Show full text]
  • Arxiv:2011.06401V1
    Noncommutative integration of the Dirac equation in homogeneous spaces A. I. Breev∗ Department of Theoretical Physics, Tomsk State University Novosobornaya Sq. 1, Tomsk, Russia, 634050 A. V. Shapovalov† Department of Theoretical Physics, Tomsk State University Novosobornaya Sq. 1, Tomsk, Russia, 634050 and Tomsk Polytechnic University, Lenin ave., 30, Tomsk, Russia, 634034 Abstract We develop a noncommutative integration method for the Dirac equation in homogeneous spaces. The Dirac equation with an invariant metric is shown to be equivalent to a system of equations on a Lie group of transformations of a homogeneous space. This allows us to effectively apply the noncommutative integration method of linear partial differential equations on Lie groups. This method differs from the well-known method of separation of variables and to some extent can often supplement it. The general structure of the method developed is illustrated with an example of a homogeneous space which does not admit separation of variables in the Dirac equation. However, the basis of exact solutions to the Dirac equation is constructed explicitly by the noncommutative integration method. Also, we construct a complete set of new exact solutions to the Dirac equation in the three-dimensional de Sitter space-time AdS3 using the method developed. The solutions arXiv:2011.06401v1 [math-ph] 12 Nov 2020 obtained are found in terms of elementary functions, which is characteristic of the noncommutative integration method. PACS numbers: 02.20.Qs, 03.65.Pm, 31.15.xh Keywords: Dirac equation; noncommutative integration; homogeneous spaces; induced representations; orbit method Mathematics Subject Classification 2010: 35Q41, 17B08, 58J70 ∗ [email protected] † [email protected] 1 I.
    [Show full text]
  • An Orbit Model for the Spectra of Nilpotent Gelfand Pairs
    AN ORBIT MODEL FOR THE SPECTRA OF NILPOTENT GELFAND PAIRS HOLLEY FRIEDLANDER, WILLIAM GRODZICKI, WAYNE JOHNSON, GAIL RATCLIFF, ANNA ROMANOV, BENJAMIN STRASSER, BRENT WESSEL Abstract. Let N be a connected and simply connected nilpotent Lie group, and let K be a subgroup of the automorphism group of N. We say that the 1 pair (K; N) is a nilpotent Gelfand pair if LK (N) is an abelian algebra under convolution. In this document we establish a geometric model for the Gelfand spectra of nilpotent Gelfand pairs (K; N) where the K-orbits in the center of N have a one-parameter cross section and satisfy a certain non-degeneracy con- dition. More specifically, we show that the one-to-one correspondence between the set ∆(K; N) of bounded K-spherical functions on N and the set A(K; N) of K-orbits in the dual n∗ of the Lie algebra for N established in [BR08] is a homeomorphism for this class of nilpotent Gelfand pairs. This result had previously been shown for N a free group and N a Heisenberg group, and was conjectured to hold for all nilpotent Gelfand pairs in [BR08]. 1. Introduction A Gelfand pair (G; K) consists of a locally compact topological group G and a compact subgroup K ⊂ G such that the space L1(G==K) of integrable K-bi- invariant functions on G is commutative. Such pairs arise naturally in harmonic analysis and representation theory of Lie groups, with perhaps the best known examples emerging in the case of a connected semisimple Lie group G with finite center and maximal compact subgroup K.
    [Show full text]
  • Lectures on the Orbit Method
    Lectures on the Orbit Method A. A. Kirillov Graduate Studies in Mathematics Volume 64 American Mathematical Society Providence, Rhode Island Contents Preface xv Introduction xvii Chapter 1. Geometry of Coadjoint Orbits 1 §1. Basic definitions 1 1.1. Coadjoint representation 1 1.2. Canonical form GQ 4 §2. Symplectic structure on coadjoint orbits 5 2.1. The first (original) approach 6 2.2. The second (Poisson) approach 7 2.3. The third (symplectic reduction) approach 9 2.4. Integrality condition 11 §3. Coadjoint invariant functions 14 3.1. General properties of invariants 14 3.2. Examples 15 §4. The moment map 16 4.1. The universal property of coadjoint orbits 16 4.2. Some particular cases 19 §5. Polarizations 23 5.1. Elements of symplectic geometry 23 5.2. Invariant polarizations on homogeneous symplectic man- ifolds 26 vn viii Contents Chapter 2. Representations and Orbits of the Heisenberg Group 31 §1. Heisenberg Lie algebra and Heisenberg Lie group 32 1.1. Some realizations 32 1.2. Universal enveloping algebra U(t)) 35 1.3. The Heisenberg Lie algebra as a contraction 37 §2. Canonical commutation relations 39 2.1. Creation and annihilation operators 39 2.2. Two-sided ideals in C/(h) 41 2.3. H. Weyl reformulation of CCR 41 2.4. The standard realization of CCR 43 2.5. Other realizations of CCR 45 2.6. Uniqueness theorem 49 §3. Representation theory for the Heisenberg group 57 3.1. The unitary dual H 57 3.2. The generalized characters of H 59 3.3. The infinitesimal characters of H 60 3.4.
    [Show full text]