ISOTROPY REPRESENTATION FOR HARISH-CHANDRA MODULES

2005 ◽  
Author(s):  
HIROSHI YAMASHITA
Keyword(s):  
Isotropy Representation

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

scholarly journals HOMOGENEOUS EINSTEIN METRICS ON GENERALIZED FLAG MANIFOLDS WITH FIVE ISOTROPY SUMMANDS

2013 ◽  
Vol 24 (10) ◽  
pp. 1350077 ◽  
Author(s):  
ANDREAS ARVANITOYEORGOS ◽  
IOANNIS CHRYSIKOS ◽  
YUSUKE SAKANE
Keyword(s):  
Einstein Equation ◽  
Einstein Metrics ◽  
Flag Manifold ◽  
Polynomial Systems ◽  
Flag Manifolds ◽  
Isotropy Representation ◽  
Generalized Flag Manifolds ◽  
A New Technique ◽  
Kähler Einstein Metrics ◽  
Homogeneous Einstein Metrics

We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad (K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Gröbner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO (2ℓ + 1)/( U (1) × U (p) × SO (2(ℓ - p - 1) + 1)) and SO (2ℓ)/( U (1) × U (p) × SO (2(ℓ - p - 1))) we prove existence of at least two non-Kähler–Einstein metrics. For small values of ℓ and p we give the precise number of invariant Einstein metrics.

ISOTROPY REPRESENTATIONS, EIGENVALUES OF A CASIMIR ELEMENT, AND COMMUTATIVE LIE SUBALGEBRAS

2001 ◽  
Vol 64 (1) ◽  
pp. 61-80 ◽  
Author(s):  
DMITRI I. PANYUSHEV
Keyword(s):  
Lie Algebra ◽  
Finite Order ◽  
Killing Form ◽  
Maximal Eigenvalue ◽  
Isotropy Representation ◽  
Semisimple Lie Algebra ◽  
Casimir Element ◽  
Root Of Unity ◽  
Exterior Powers

Let [hfr ] be a reductive subalgebra of a semisimple Lie algebra [gfr ] and C[hfr ] ∈ U([hfr ]) be the Casimir element determined by the restriction of the Killing form on [gfr ] to [hfr ]. The paper studies eigenvalues of C[hfr ] on the isotropy representation [mfr ]≃[gfr ]/[hfr ]. Some general estimates connecting the eigenvalues and the Dynkin indices of [mfr ] are given. If [hfr ] is a symmetric subalgebra, it is shown that describing the maximal eigenvalue of C[hfr ] on exterior powers of [mfr ] is connected with possible dimensions of commutative Lie subalgebras in [mfr ], thereby extending a result of Kostant. In this situation, a formula is also given for the maximal eigenvalue of C[hfr ] on ∧ [mfr ]. More generally, a similar picture arises if [hfr ] = [gfr ]Θ, where Θ is an automorphism of finite order m and [mfr ] is replaced by the eigenspace of Θ corresponding to a primitive mth root of unity.

Polar representations and symmetric spaces

1999 ◽  
Vol 1999 (507) ◽  
pp. 93-106 ◽  
Author(s):  
J.-H Eschenburg ◽  
E Heintze
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Isotropy Representation ◽  
Polar Representation ◽  
Conceptual Proof

Abstract J. Dadok has shown by classification that any polar representation has the same orbits as the isotropy representation of some symmetric space. A conceptual proof of this result is given subject to some restriction.

scholarly journals The isotropy representation for homogeneous Siegel domains

1985 ◽  
Vol 120 (2) ◽  
pp. 295-326 ◽  
Author(s):  
Joseph D’Atri ◽  
Josef Dorfmeister ◽  
Yan Da Zhao
Keyword(s):  
Isotropy Representation ◽  
Siegel Domains

On the Structure of a Class of Graded Modules Related to Symmetric Pairs

2006 ◽  
Vol 13 (02) ◽  
pp. 315-328 ◽  
Author(s):  
Gang Han
Keyword(s):  
Lie Algebra ◽  
Adjoint Representation ◽  
Exterior Algebra ◽  
Symmetric Algebra ◽  
Module Structure ◽  
Cartan Decomposition ◽  
Isotropy Representation ◽  
Semisimple Lie Algebra ◽  
Graded Modules ◽  
Special Case

Let [Formula: see text] be the Cartan decomposition of a real semisimple Lie algebra and [Formula: see text] be its complexification. Let [Formula: see text] be the corresponding isotropy representation, and the exterior algebra [Formula: see text] becomes a graded [Formula: see text]-module by extending ν. We study a graded [Formula: see text]-submodule C of [Formula: see text] and get two important decompositions of the [Formula: see text]-module [Formula: see text]. Let [Formula: see text] be the symmetric algebra over [Formula: see text]. Then [Formula: see text] also has an [Formula: see text]-module structure, which is [Formula: see text]-equivariant, and C is a space of generators for this module. Our results generalize Kostant's results in the special case that ν is the adjoint representation of a semisimple Lie algebra.

ON THE NUMBER OF INVARIANT EINSTEIN METRICS ON A COMPACT HOMOGENEOUS SPACE, NEWTON POLYTOPES AND CONTRACTIONS OF LIE ALGEBRAS

2006 ◽  
Vol 03 (05n06) ◽  
pp. 1047-1075 ◽  
Author(s):  
MICHAIL M. GRAEV
Keyword(s):  
Homogeneous Space ◽  
Lie Algebras ◽  
Einstein Metrics ◽  
Compact Convex ◽  
Convex Polytope ◽  
Invariant Metric ◽  
Isotropy Representation ◽  
Newton Polytopes ◽  
Number Formula ◽  
Proper Face

We associate to a homogeneous manifold M = G/H, with a simple spectrum of the isotropy representation, a compact convex polytope PM which is the Newton polytope of the rational function s(t) and that to each invariant metric t of M associates its scalar curvature. We estimate the number [Formula: see text] of isolate invariant holomorphic Einstein metrics (up to homothety) on Mℂ = Gℂ/Hℂ. Using the results of A. G. Kouchnirenko and D. N. Bernstein, we prove that [Formula: see text], where ν(M) is the integer volume of PM, and give conditions when the defect [Formula: see text]. In case when G is a compact semisimple Lie group, the positiveness of d(M) is related with the existence of Ricci-flat holomorphic metric on a complexification of a noncompact homogeneous space Mγ = Gγ/HP which is a contraction of M and is associated with a proper face γ of PM.

scholarly journals New homogeneous Einstein metrics on quaternionic Stiefel manifolds

2018 ◽  
Vol 18 (4) ◽  
pp. 509-524 ◽  
Author(s):  
Andreas Arvanitoyeorgos ◽  
Yusuke Sakane ◽  
Marina Statha
Keyword(s):  
Homogeneous Space ◽  
Einstein Metrics ◽  
Flag Manifold ◽  
Total Space ◽  
Stiefel Manifold ◽  
Isotropy Representation ◽  
Stiefel Manifolds ◽  
Generalized Flag Manifold ◽  
Generalized Wallach Space ◽  
Homogeneous Einstein Metrics

Abstract We consider invariant Einstein metrics on the quaternionic Stiefel manifold Vpℍn of all orthonormal p-frames in ℍn. This manifold is diffeomorphic to the homogeneous space Sp(n)/Sp(n − p) and its isotropy representation contains equivalent summands. We obtain new Einstein metrics on Vpℍn ≅ Sp(n)/Sp(n − p), where n = k1 + k2 + k3 and p = n − k3. We view Vpℍn as a total space over the generalized Wallach space Sp(n)/(Sp(k1)×Sp(k2)×Sp(k3)) and over the generalized flag manifold Sp(n)/(U(p)×Sp(n − p)).

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