Computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces

In this paper we introduce the history and present situation of the computation of the cohomology rings of Kac-Moody groups,their flag manifolds and classifying spaces,and give some problems and conjectures that deserve further study.

The Signature of Generalized Flag Manifolds and Applications

Let G be a linear algebraic group over C and P be a parabolic subgroup. We determine the signatures of the flag manifold G/P. As an application, we prove that the nonsingular hypersurfaces of degree 2 in CP^n are prime if n satisfies certain conditions.

The symmetric space,strong isotropy irreducibility and equigeodesic properties

A smooth curve on a homogeneous manifold G/H is called a Riemannian equigeodesic if it is a homogeneous geodesic for any G-invariant Riemannian metric.The homogeneous manifold G/H is called Riemannian equigeodesic,if for any x∈G/H and any nonzero y∈Tx(G/H),there exists a Riemannian equigeodesic c(t) with c(0)=x and ■(0)=y.These two notions can be naturally transferred to the Finsler setting,which provides the definitions for Finsler equigeodesics and Finsler equigeodesic spaces.We prove two classification theorems for Riemannian equigeodesic spaces and Finsler equigeodesic spaces,respectively.Firstly,a homogeneous manifold G/H with a connected simply connected quasi compact G and a connected H is Riemannian equigeodesic if and only if it can be decomposed as a product of Euclidean factors and compact strongly isotropy irreducible factors.Secondly,a homogeneous manifold G/H with a compact semisimple G is Finsler equigeodesic if and only if it can be locally decomposed as a product,in which each factor is Spin(7)/G2,G2/SU (3) or a symmetric space of compact type.These results imply that the symmetric space and the strongly isotropy irreducible space of compact type can be interpreted by equigeodesic properties.As an application,we classify the homogeneous manifold G/H with a compact semisimple G such that all the G-invariant Finsler metrics on G/H are Berwald.It suggests a new project in homogeneous Finsler geometry,i.e.,to systematically study the homogeneous manifold G/H on which all the G-invariant Finsler metrics satisfy a certain geometric property.

Maps from a Simply Connected Space to Flag Manifold G/T

In this paper the classification of maps from a simply connected space X to a flag manifold G/T is studied.As an application,the structure of the homotopy set for self-maps of flag manifolds is determined.

On the Kernel of the Borel’s Characteristic Map of Lie Groups

For compact and connected Lie group G with a maximal torus T the quotient space G/T is canonically a smooth projective manifold,known as the complete flag manifold of the group G.The cohomology ring map c^(∗):H^(∗)(B T)→H∗(G/T)induced by the inclusion c:G/T→B_(T) is called the Borel’s characteristic map of the group G[7,8],where B T denotes the classifying space of T.Let G be simply-connected and simple.Based on the Schubert presentation of the cohomology H^(∗)(G/T)of the flag manifold G/T obtained in[10,11],we develop a method to find a basic set of explicit generators for the kernel ker c^(∗)⊂H^(∗)(B_(T))of the characteristic map c.

New Homogeneous Einstein Metrics on SO(7)/T

The authors compute non-zero structure constants of the full flag manifold M = SO(7)/T with nine isotropy summands,then construct the Einstein equations.With the help of computer they get all the forty-eight positive solutions(up to a scale) for SO(7)/T,up to isometry there are only five G-invariant Einstein metrics,of which one is Khler Einstein metric and four are non-Khler Einstein metrics.