GEOMETRY OF FLAG MANIFOLDS

A generalized metric on a manifold M M , i.e., a pair ( g , H ) (g,H) , where g g is a Riemannian metric and H H a closed 3 3 -form, is a fixed point of the generalized Ricci flow if and only if ( g , H ) (g,H) is Bismut Ricci flat: H H is g g -harmonic and R c ( g ) = 1 4 H g 2 Rc(g)=\tfrac {1}{4}H_g^2 . On any homogeneous space M = G / K M=G/K , where G = G 1 × G 2 G=G_1\times G_2 is a compact semisimple Lie group with two simple factors, under some mild assumptions, we exhibit a Bismut Ricci flat G G -invariant generalized metric, which is proved to be unique among a 4 4 -parameter space of metrics in many cases, including when K K is neither abelian nor semisimple. On the other hand, if K K is simple and the standard metric is Einstein on both G 1 / π 1 ( K ) G_1/\pi _1(K) and G 2 / π 2 ( K ) G_2/\pi _2(K) , we give a one-parameter family of Bismut Ricci flat G G -invariant generalized metrics on G / K G/K and show that it is most likely pairwise non-homothetic by computing the ratio of Ricci eigenvalues. This is proved to be the case for every space of the form M = G × G / Δ K M=G\times G/\Delta K and for M 35 = S O ( 8 ) × S O ( 7 ) / G 2 M^{35}=SO(8)\times SO(7)/G_2 .

Riemannian g.o. metrics in certain M-spaces

Let (E, h) be a semipositive hermitian holomorphic line bundle on a compact complex manifold X with dimX>11$$\\end{document}]]>. Assume that for each point x∈X there exists a divisor D∈|E| in the complete linear system determined by E whose complement X\\D is a Stein neighbourhood of x with the density property. Then, the disc bundle Δh(E)={e∈E:|e|h<1} is an Oka manifold while Dh(E)={e∈E:|e|h>1}1\\}$$\\end{document}]]> is a Kobayashi hyperbolic domain. In particular, the zero section of E admits a basis of Oka neighbourhoods {|e|h<c} with c>00$$\\end{document}]]>. We show that this holds if X is a rational homogeneous manifold of dimension >11$$\\end{document}]]>. This class of manifolds includes complex projective spaces, Grassmannians, and flag manifolds. This phenomenon contributes to the heuristic principle that Oka properties are related to metric positivity of complex manifolds.

We prove that any invariant strong Kähler structure with torsion (SKT structure) on a flag manifold M = G/K of a semi-simple compact Lie group G is Kähler. As an application we describe invariant generalized Kähler structures on M.

The geometric quantization on a homogeneous manifold is studied. For any quantizable function f , the stochastical expression for the unitary group exp(i tQ λ(f)) generated by the quantized operator Q λ( f ) is established. As an application, a Feynman-Kac formula for the compact semisimple Lie group is rederived.

Let G be a compact semisimple Lie group, g its Lie algebra, (,) an AdG-invariant inner product on g, and M an adjoint orbit in 9. In this article, if (M,(,)|M) is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold L⊂M, upper bounds on the first positive eigenvalue λ1(L) of the Laplacian ΔL, which acts on C∞(L), and lower bounds on the volume of L. In particular, when (M,(,)|M) is Kähler-Einstein, (p=cω, where p and ω are Ricci form and Kähler form of (M,(,)|M) with respect to the canonical complex structure respectively, and c is a positive constant,) we prove λ1(L)≤c. Combining with a result of Oh [5], we can see that L is Hamiltonian stable if and only if λ1(L)=c.

Reductive homogeneous Lorentzian manifolds

The Bott–Borel–Weil theorem (BBW) is a statement about a complex homogeneous space X=G/R where G is a compact semisimple Lie group and R is the centralizer of a torus in G. One knows that BBW is equivalent to the determination of how R operates on the cohomology of a certain nilpotent Lie algebra of antiholomorphic tangent vectors operating on an arbitrary irreducible G-module. Upon replacing the complex structure with a space S of spinors BBW is equivalent to a statement about the eigenvalues of the Casimir operator of R in the module S ⊗ Vλ where Vλ is the irreducible G-module with highest weight λ. But then the complex structure is eliminated and the problem makes sense for any compact subgroup R ⊂ G. We solve the problem in the case where R and G have the same rank. The very special case where R is the centralizer of a torus then yields BBW. Involved in the general case (i.e. arbitrary R) is a new (at least to mathematicians) Dirac operator with a cubic term whose square is expressed in terms of the Casimir operator. Actually the spectral resolution of the Dirac operator is determined in the general case. The kernel of the Dirac operator is only given in the equal rank case. A key feature in that case is that, associated to λ, there is a multiplet of representations of R, having rather striking properties, and cardinality equal to the Euler number of G/R.

Let [Formula: see text] be a compact Lie group acting on a smooth manifold [Formula: see text]. In this paper, we consider Meinrenken’s [Formula: see text]-equivariant bundle gerbe connections on [Formula: see text] as objects in a 2-groupoid. We prove this 2-category is equivalent to the 2-groupoid of gerbe connections on the differential quotient stack associated to [Formula: see text], and isomorphism classes of [Formula: see text]-equivariant gerbe connections are classified by degree 3 differential equivariant cohomology. Finally, we consider the existence and uniqueness of conjugation-equivariant gerbe connections on compact semisimple Lie groups.

Stability results for approximation by positive definite functions on SO(3)

Let G be a connected, complex reductive group and B, a Borel subgroup. The homogeneous space G/B is called the (full) flag variety of G; for GL n this is the ordinary variety of flags of subspaces in ℂ n . The properties of this variety have many implications in the study of reductive groups and their representations. For example, all the irreducible finite dimensional G-modules can be obtained as global sections of line bundles on the flag variety. More generally, any of its (G-linearized) line bundles has at most one non-vanishing sheaf-cohomology space, having a natural structure of a simple G-module if G is simply connected. This is part of the content of the Borel-Weil-Bott theorem and can be seen as a geometric interpretation of E. Cartan’s theory of highest weights.

One describes, using a detailed analysis of Atiyah-Hirzebruch spectral sequence, the tuples of cohomology classes on a compact, complex manifold, corresponding to the Chern classes of a complex vector bundle of stable rank. This classification becomes more effective on generalized flag manifolds, where the Lie algebra formalism and concrete integrability conditions describe in constructive terms the Chern classes of a vector bundle.

Sharp Fourier type and cotype of Lebesgue spaces and Schatten classes with respect to an arbitrary compact semisimple Lie group are investigated. In the process, a local variant of the Hausdorff-Young inequality on such groups is given.

Functions generated by blocks were introduced by M. Taibleson and G. Weiss in the setting of the one-dimensional torus T T [TW1]. They showed that these functions formed a space "close" to the class of integrable functions for which we have almost everywhere convergence of Fourier series. Together with S. Lu [LTW] they extended the theory to the n n -dimensional torus where this convergence result (for Bochner-Riesz means at the critical index) is valid provided we also restrict ourselves to L log ⁡ L L\log L . In this paper we show that this restriction is not needed if the underlying domain is a compact semisimple Lie group (or certain more general spaces of a homogeneous type). Other considerations (for example, these spaces form an interesting family of quasi-Banach spaces; they are connected with the notion of entropy) guide one in their study. We show how this point of view can be exploited in the setting of more general underlying domains.

Pluriclosed metrics on compact semisimple Lie groups