A generalized Cartan decomposition for the double coset space $(U(n_1) \times U(n_2) \times U(n_3)) \backslash U(n) / (U(p) \times U(q))$

AbstractLet G/K be a noncompact symmetric space, Gc/K its compact dual, 𝔤=𝔨⊕𝔭 the Cartan decomposition of the Lie algebra 𝔤 of G, 𝔞 a maximal abelian subspace of 𝔭, H be an element of 𝔞, a=exp (H) , and ac =exp (iH) . In this paper, we prove that if for some positive integer r, νrac is absolutely continuous with respect to the Haar measure on Gc, then νra is absolutely continuous with respect to the left Haar measure on G, where νac (respectively νa) is the K-bi-invariant orbital measure supported on the double coset KacK (respectively KaK). We also generalize a result of Gupta and Hare [‘Singular dichotomy for orbital measures on complex groups’, Boll. Unione Mat. Ital. (9) III (2010), 409–419] to general noncompact symmetric spaces and transfer many of their results from compact symmetric spaces to their dual noncompact symmetric spaces.

Download Full-text

Elementary Proof of the Fundamental Lemma For a Unitary Group

Canadian Journal of Mathematics ◽

10.4153/cjm-1998-005-4 ◽

1998 ◽

Vol 50 (1) ◽

pp. 74-98 ◽

Cited By ~ 1

Author(s):

Yuval Z. Flicker

Keyword(s):

Unitary Group ◽

Compact Subgroup ◽

Double Coset ◽

Base Change ◽

Coset Space ◽

Fundamental Lemma ◽

Regular Elements ◽

Orbital Integral ◽

Single Orbit ◽

Endoscopic Group

AbstractThe fundamental lemma in the theory of automorphic forms is proven for the (quasi-split) unitary group U(3) in three variables associatedwith a quadratic extension of p-adic fields, and its endoscopic group U(2), by means of a new, elementary technique. This lemma is a prerequisite for an application of the trace formula to classify the automorphic and admissible representations of U(3) in terms of those of U(2) and base change to GL(3). It compares the (unstable) orbital integral of the characteristic function of the standard maximal compact subgroup K of U(3) at a regular element (whose centralizer T is a torus), with an analogous (stable) orbital integral on the endoscopic group U(2). The technique is based on computing the sum over the double coset space T\G/K which describes the integral, by means of an intermediate double coset space H\G/K for a subgroup H of G= U(3) containing T. Such an argument originates from Weissauer's work on the symplectic group. The lemma is proven for both ramified and unramified regular elements, for which endoscopy occurs (the stable conjugacy class is not a single orbit).

Download Full-text

GROUP THEORETIC FORMULATION OF THE SEGAL-WILSON APPROACH TO INTEGRABLE SYSTEMS WITH APPLICATIONS

Reviews in Mathematical Physics ◽

10.1142/s0129055x92000121 ◽

1992 ◽

Vol 04 (03) ◽

pp. 451-499 ◽

Cited By ~ 11

Author(s):

G. HAAK ◽

M. SCHMIDT ◽

R. SCHRADER

Keyword(s):

Integrable Systems ◽

Equations Of Motion ◽

Double Coset ◽

Scattering Data ◽

Theoretic Approach ◽

Coset Space ◽

Integrals Of Motion ◽

Natural Group ◽

Left Multiplication ◽

Starting Point

A general group theoretic formulation of integrable systems is presented. The approach generalizes the discussion of the KdV equations of Segal and Wilson based on ideas of Sato. The starting point is the construction of commuting flows on the group via left multiplication with elements from an abelian subgroup. The initial data are then coded by elements, called abstract scattering data, in a certain coset space. The resulting equations of motion are then derived from a suitably formulated Maurer-Cartan equation (zero curvature condition) given an abstract Birkhoff factorization. The resulting equations of motion are of the Zakharov-Shabat type. In the case of flows periodic in x-space, the integrals of motion have a natural group theoretic interpretation. A first example is provided by the generalized nonlinear Schrödinger equation, first studied by Fordy and Kulish with integrals of motion which may be local or nonlocal. A suitable reduction gives the mKdV equations of Drinfeld and Sokolov. On the level of abstract scattering data the generalized Miura transformation from solutions of the mKdV equations to the KdV type equations is then just the canonical map from a coset space to a double coset space. This group theoretic approach is related to the algebraic geometric discussion of integrable systems via an affine map from the abelian group describing flows restricted to a suitable set of abstract scattering data, called algebraic geometric, onto a connected component of the Picard variety.

Download Full-text

VISIBLE ACTIONS ON FLAG VARIETIES OF TYPE B AND A GENERALISATION OF THE CARTAN DECOMPOSITION

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972712000615 ◽

2012 ◽

Vol 88 (1) ◽

pp. 81-97 ◽

Cited By ~ 1

Author(s):

YUICHIRO TANAKA

Keyword(s):

Double Coset ◽

Type A ◽

Levi Subgroup ◽

Flag Varieties ◽

Type B ◽

Compact Lie Groups ◽

Cartan Decomposition ◽

Sigma H ◽

Visible Action ◽

Multiplicity Free

AbstractWe give a generalisation of the Cartan decomposition for connected compact Lie groups of type B motivated by the work on visible actions of Kobayashi [‘A generalized Cartan decomposition for the double coset space $(U(n_{1})\times U(n_{2})\times U(n_{3})) \backslash U(n)/ (U(p)\times U(q))$’, J. Math. Soc. Japan59 (2007), 669–691] for type A groups. Suppose that $G$ is a connected compact Lie group of type B, $\sigma $ is a Chevalley–Weyl involution and $L$, $H$ are Levi subgroups. First, we prove that $G=LG^{\sigma }H$ holds if and only if either (I) both $H$ and $L$ are maximal and of type A, or (II) $(G,H)$ is symmetric and $L$ is the Levi subgroup of an arbitrary maximal parabolic subgroup up to switching $H$ and $L$. This classification gives a visible action of $L$ on the generalised flag variety $G/H$, as well as that of the $H$-action on $G/L$ and of the $G$-action on $(G\times G)/(L\times H)$. Second, we find an explicit ‘slice’ $B$ with $\dim B=\mathrm {rank}\, G$ in case I, and $\dim B=2$ or $3$ in case II, such that a generalised Cartan decomposition $G=LBH$holds. An application to multiplicity-free theorems of representations is also discussed.

Download Full-text

Invariant measures on double coset spaces

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700028524 ◽

1965 ◽

Vol 5 (4) ◽

pp. 495-505 ◽

Cited By ~ 4

Author(s):

Teng-Sun Liu

Keyword(s):

Compact Group ◽

Closed Subgroup ◽

Invariant Measures ◽

Double Coset ◽

Locally Compact Group ◽

Coset Space ◽

Canonical Mapping ◽

Locally Compact ◽

Coset Spaces ◽

Image Position

Let G be a locally compact group with left invariant Haar measure m. Le H be a closed subgroup of G and K a compact group of G. Let R be the equivalence relation in G defined by (a, b)∈R if and if a = kbh for some k in K and h in H. We call E =G/R the double coset space of G modulo K and H. Donote by a the canonical mapping of G onto E. It can be shown that E is a locally compact space and α is continous and open Let N be the normalizer of K in G, i. e. .

Download Full-text

Symmetry and unification from soft theorems and unitarity

Journal of High Energy Physics ◽

10.1007/jhep05(2021)161 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Clifford Cheung ◽

Zander Moss

Keyword(s):

Symmetry Algebra ◽

Scalar Fields ◽

Particle Interactions ◽

Massive Scalar ◽

Coset Space ◽

Dimensional Theory ◽

Finite Spectrum ◽

Physical Constraints ◽

High Energies ◽

Zero Condition

Abstract We argue that symmetry and unification can emerge as byproducts of certain physical constraints on dynamical scattering. To accomplish this we parameterize a general Lorentz invariant, four-dimensional theory of massless and massive scalar fields coupled via arbitrary local interactions. Assuming perturbative unitarity and an Adler zero condition, we prove that any finite spectrum of massless and massive modes will necessarily unify at high energies into multiplets of a linearized symmetry. Certain generators of the symmetry algebra can be derived explicitly in terms of the spectrum and three-particle interactions. Furthermore, our assumptions imply that the coset space is symmetric.

Download Full-text

Discrete symmetries and coset space dimensional reduction

Physics Letters B ◽

10.1016/0370-2693(89)90565-0 ◽

1989 ◽

Vol 232 (1) ◽

pp. 104-112 ◽

Cited By ~ 11

Author(s):

D. Kapetanakis ◽

G. Zoupanos

Keyword(s):

Dimensional Reduction ◽

Discrete Symmetries ◽

Coset Space

Download Full-text

LOCAL FRACTIONAL SUPERSYMMETRY FOR ALTERNATIVE STATISTICS

Modern Physics Letters A ◽

10.1142/s0217732396000916 ◽

1996 ◽

Vol 11 (11) ◽

pp. 899-913 ◽

Cited By ~ 26

Author(s):

N. FLEURY ◽

M. RAUSCH DE TRAUBENBERG

Keyword(s):

Group Theory ◽

Braid Group ◽

Equation Of Motion ◽

Coset Space ◽

One Dimensional ◽

The One

A group theory justification of one-dimensional fractional supersymmetry is proposed using an analog of a coset space, just like the one introduced in 1-D supersymmetry. This theory is then gauged to obtain a local fractional supersymmetry, i.e. a fractional supergravity which is then quantized à la Dirac to obtain an equation of motion for a particle which is in a representation of the braid group and should describe alternative statistics. A formulation invariant under general reparametrization is given by means of a curved fractional superline.

Download Full-text