Dense orbits for parabolic group actions and fans.

We consider actions of parabolic subgroups in a general linear group on ideals in the Lie algebra of the unipotent radical. It is convenient to fix the shape of the ideal and the number t of blocks of the parabolic group. Then we consider all those parabolic groups P(d) (where d=(d_1,...,d_t) denotes the block size) acting on the corresponding ideal n(d) for all possible dimension vectors d simultaneously. We define the set D(n) to be the set of all d, so that P(d) acts with a dense orbit on n(d). If n is the unipotent radical, then D(n) consists of all dimension vectors, similarly, if P(d) acts with a finite number of orbits for all d.

The main result of my talk defines a stratification of the set D(n) into t-dimensional cones. The set D(n) becomes the set of the lattice points of a smooth t-dimensional fan and the dense orbits for two lattice points in the same cone of the fan have an analogous representative.