Projective subspaces in the variety of normal sections and tangent spaces to a symmetric space (original) (raw)
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Geometriae Dedicata, 1999
We continue the study of the variety X[M ] of planar normal sections on a natural embedding of a flag manifold M. Here we consider those subvarieties of X[M ] that are projective spaces. When M = G/T is the manifold of complete flags of a compact simple Lie group G, we obtain our main results. The first one characterizes those subspaces of the tangent space T [T ] (M), invariant by the torus action and which give rise to real projective spaces in X[M ]. The other one is the following. Let p be the tangent space of the inner symmetric space G/K at [K]. Then RP (p) is maximal in X[M ] if and only if π2(G/K) does not vanish.
ON THE VARIETY OF PLANAR NORMAL SECTIONS
In the present paper we present a survey of results concerning the variety X [M ] of planar normal sections associated to a natural embedding of a real flag manifold M m . The results included are those that, we feel, better describe the nature of this algebraic variety of RP m−1 . In particular we present results concerning its Euler characteristic showing that it depends only on dim M and not on the nature of M itself. Furthermore, when M is the manifold of complete flags of a compact simple Lie group, we present what is, in some sense, its dimension and a large class of submanifolds of RP m−1 contained in X [M ].
The Student Mathematical Library, 2003
Let G be a complex reductive linear algebraic group and G0 G a real form. Suppose P is a parabolic subgroup of G and assume that P has a Levi factor L such that G0 \ L = L0 is a real form of L. Using the minimal globalization Vmin of a finite length admissible representation for L0 , one can define a homogeneous analytic vector bundle on the G0 orbit S of P in the generalized flag manifold Y = G=P. Let A(P;Vmin) denote the corresponding sheaf of polarized sections. In this article we analyze the G0 representations obtained on the compactly supported sheaf cohomology groups H p c (S; A(P;Vmin)).
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Let P be a parabolic subgroup of a complex simple linear algebraic group G. We prove that the tangent bundle T (G/P) is stable.
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Canadian Journal of Mathematics, 2000
We study K-orbits in G/P where G is a complex connected reductive group, P ⊆ G is a parabolic subgroup, and K ⊆ G is the fixed point subgroup of an involutive automorphism θ. Generalizing work of Springer, we parametrize the (finite) orbit set K \ G/P and we determine the isotropy groups. As a consequence, we describe the closed (resp. affine) orbits in terms of θ-stable (resp. θ-split) parabolic subgroups. We also describe the decomposition of any (K, P)-double coset in G into (K, B)-double cosets, where B ⊆ P is a Borel subgroup. Finally, for certain K-orbit closures X ⊆ G/B, and for any homogeneous line bundle on G/B having nonzero global sections, we show that the restriction map resX : H 0(G/B, ) → H 0(X, ) is surjective and that Hi (X, ) = 0 for i ≥ 1. Moreover, we describe the K-module H 0(X, ). This gives information on the restriction to K of the simple G-module H 0(G/B, ). Our construction is a geometric analogue of Vogan and Sepanski’s approach to extremal K-types.
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Geometriae Dedicata, 1997
This paper contains a proof of the following property of compact irreducible Hermitian symmetric spaces. If H=G/K where G is a compact simply connected simple Lie group, T a maximal torus of G and F(T,H)=|E 1,...,E m is the fixed point set of T on H, then for each pair E i , E j there is a 2-dimentional sphere N ij ⊂ H such that E i and E j are antipodal points of N ij.