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56
Riemannian flag manifolds with homogeneous geodesics
 TRANS. AMER. MATH. SOC
, 2007
"... A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a oneparameter subgroup of the Lie group G. We investigate Ginvariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifo ..."
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Cited by 13 (1 self)
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A geodesic in a Riemannian homogeneous manifold (M = G/K, g) is called a homogeneous geodesic if it is an orbit of a oneparameter subgroup of the Lie group G. We investigate Ginvariant metrics with homogeneous geodesics (i.e., such that all geodesics are homogeneous) when M = G/K is a flag manifold, that is, an adjoint orbit of a compact semisimple Lie group G. We use an important invariant of a flag manifold M = G/K, itsTroot system, to give a simple necessary condition that M admits a nonstandard Ginvariant metric with homogeneous geodesics. Hence, the problem reduces substantially to the study of a short list of prospective flag manifolds. A common feature of these spaces is that their isotropy representation has two irreducible components. We prove that among all flag manifolds M = G/K of a simple Lie group G, only the manifold Com(R2ℓ+2)=SO(2ℓ +1)/U(ℓ) of complex structures in R2ℓ+2, and the complex projective space CP 2ℓ−1 = Sp(ℓ)/U(1) · Sp(ℓ − 1) admit a nonnaturally reductive invariant metric with homogeneous geodesics. In all other cases the only Ginvariant metric with homogeneous geodesics is the metric which is homothetic to the standard metric (i.e., the metric associated to the negative of the Killing form of the Lie algebra g of G). According to F. Podestà and G.Thorbergsson (2003), these manifolds are the only nonHermitian symmetric flag manifolds with coisotropic action of the stabilizer.
Matroids and geometric invariant theory of torus actions on flag spaces
, 2005
"... Abstract. Let F//T be a G.I.T. quotient of a flag manifold F by the natural action of the maximal torus T in SL(n, C). The construction of the quotient space depends upon the choice of a Tlinearized line bundle L of F. We study the case where L = Lλ is a very ample homogeneous line bundle determine ..."
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Cited by 13 (3 self)
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Abstract. Let F//T be a G.I.T. quotient of a flag manifold F by the natural action of the maximal torus T in SL(n, C). The construction of the quotient space depends upon the choice of a Tlinearized line bundle L of F. We study the case where L = Lλ is a very ample homogeneous line bundle determined by a dominant weight λ. We apply a theorem of Gel’fand, Goresky, MacPherson, and Serganova about matroids and their respective polytopes to study semistability of flags relative to a given Tlinearization of Lλ. The main theorem of this note is that regardless of the choice of linearization, the semistable flags are detected by invariant sections of Lλ; that is, for each semistable flag p ∈ F, there is a Tinvariant section s of Lλ such that s(p) ̸ = 0. Additionally we find that the closure of any Torbit in F is projectively normal for any projective embedding of F.
Generalized Chebyshev Polynomials Associated with Affine Weyl Groups
 Trans. Amer. Math. Soc
, 1988
"... ABSTRACT. We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in onetoone correspondence with affine Weyl groups. For each such figure in ndimensional Euclidean space, we construct a sequence of polynomi ..."
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Cited by 12 (0 self)
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ABSTRACT. We begin with a compact figure that can be folded into smaller replicas of itself, such as the interval or equilateral triangle. Such figures are in onetoone correspondence with affine Weyl groups. For each such figure in ndimensional Euclidean space, we construct a sequence of polynomials P/c: Rn — ► Rn so that the mapping P ^ is conjugate to stretching the figure by a factor A; and folding it back onto itself. If re = 1 and the figure is the interval, this construction yields the Chebyshev polynomials (up to conjugation). The polynomials Pk are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials.
Zetafunctions of root systems
 in “Proceedings of the Conference on Lfunctions
"... Abstract. In this paper, we introduce multivariable zetafunctions of roots, and prove the analytic continuation of them. For the root systems associated with Lie algebras, these functions are also called Witten zetafunctions associated with Lie algebras which can be regarded as several variable g ..."
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Cited by 12 (9 self)
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Abstract. In this paper, we introduce multivariable zetafunctions of roots, and prove the analytic continuation of them. For the root systems associated with Lie algebras, these functions are also called Witten zetafunctions associated with Lie algebras which can be regarded as several variable generalizations of Witten zetafunctions defined by Zagier. In the case of type Ar, we have already studied some analytic properties in our previous paper. In the present paper, we prove certain functional relations among these functions of types Ar (r = 1, 2, 3) which include what is called Witten’s volume formulas. Moreover we mention some structural background of the theory of functional relations in terms of Weyl groups. 1.
LieAlgebraic Conditions for Exponential Stability of Switched Systems
 Proc. 38th Conf. on Decision and Control
, 1999
"... It has recently been shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which means that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we p ..."
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Cited by 10 (2 self)
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It has recently been shown that a family of exponentially stable linear systems whose matrices generate a solvable Lie algebra possesses a quadratic common Lyapunov function, which means that the corresponding switched linear system is exponentially stable for arbitrary switching. In this paper we prove that the same properties hold under the weaker condition that the Lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact Lie group. The corresponding local stability result for nonlinear switched systems is also established. Moreover, we demonstrate that if a Lie algebra fails to satisfy the above condition, then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable. Relevant facts from the theory of Lie algebras are collected at the end of the paper for easy reference. 1 Introduction A switched system can be described by a family of continuoustime subsystem...
Paving Hessenberg varieties by affines
, 2004
"... Abstract. Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety which arise in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all ..."
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Cited by 8 (3 self)
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Abstract. Regular nilpotent Hessenberg varieties form a family of subvarieties of the flag variety which arise in the study of quantum cohomology, geometric representation theory, and numerical analysis. In this paper we construct a paving by affines of regular nilpotent Hessenberg varieties for all classical types. This paving is in fact the intersection of a particular Bruhat decomposition with the Hessenberg variety. The nonempty cells of this paving and their dimensions can be identified by a combinatorial condition on roots. We use this paving to prove these Hessenberg varieties have no odddimensional homology. 1.
INVARIANT EINSTEIN METRICS ON FLAG MANIFOLDS WITH FOUR ISOTROPY SUMMANDS
, 904
"... Abstract. A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands and we study their geometry. We present new Ginvariant Einstein metrics by so ..."
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Abstract. A generalized flag manifold is a homogeneous space of the form G/K, where K is the centralizer of a torus in a compact connected semisimple Lie group G. We classify all flag manifolds with four isotropy summands and we study their geometry. We present new Ginvariant Einstein metrics by solving explicity the Einstein equation. We also examine the isometric problem for these Einstein metrics. 2000 Mathematics Subject Classification. Primary 53C25; Secondary 53C30.
Zeghib Actions of noncompact semisimple groups on Lorentz manifolds, Geom
 Funct. Anal
"... Abstract. The above title is the same, but with “semisimple ” instead of “simple, ” as that of a notice by Nadine Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half ..."
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Abstract. The above title is the same, but with “semisimple ” instead of “simple, ” as that of a notice by Nadine Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half of the announced results before her premature death. Here, using a different, geometric approach, we generalize her results to the semisimple case, and give proofs of all her announced results. 1.
Convex functions on symmetric spaces and geometric invariant theory for spaces of weighted configurations on flag manifolds
, 2004
"... ..."
The structure of quantum Lie algebra for the classical series Bl
 Cl and Dl, J. Phys. A: Math. Gen
, 1998
"... Abstract. The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum ClebschGordan coeffic ..."
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Abstract. The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum ClebschGordan coefficients for adjoint ⊗ adjoint → adjoint. We present a practical method for the determination of these quantum ClebschGordan coefficients and are thus able to give explicit expressions for the structure constants of the quantum Lie algebras associated to the classical Lie algebras Bl, Cl and Dl. In the quantum case also the structure constants of the Cartan subalgebra are nonzero and we observe that they are determined in terms of the simple quantum roots. We introduce an invariant Killing form on the quantum Lie algebras and find that it takes values which are simple qdeformations of the classical ones. 1.