Computation of the cohomology rings of Kac-Moody groups, their flag manifolds and classifying spaces (Q2141330)

This is a survey article, which concerns the Kac-Moody groups. These are simply connected Lie groups associated to (generalized) Cartan matrices. A Cartan matrix is \textit{indecomposable} if it cannot be written as a direct sum of two Cartan matrices. An \textit{indecomposable} Kac-Moody group is one that is associated to an indecomposable Cartan matrix. There are three types of indecomposable Kac-Moody groups: finite, affine and indefinite. The finite type includes some classes of classical Lie groups and exceptional Lie groups. This paper actually deals with the calculation of cohomology rings of Kac-Moody groups, their flag manifolds and their classifying spaces, with integer, rational and mod \(p\) coefficients (where \(p\) is a prime). The author presents some historical background and current developments in the topic. The main tool in these computations has been various kinds of spectral sequences. The paper ends with a set of problems and conjectures concerning these cohomology rings.