Equivariant compactifications of reductive groups

Under study are equivariant projective compactifications of reductive groups that can be obtained as the closure of the image of the group in the space of projective linear operators of a representation. The structure and the mutual position of the orbits of the action of the direct square of the group acting by left/right multiplication and the local structure of the compactification in the neighbourhood of a closed orbit are described. Several conditions for the normality and smoothness of a compactification are obtained. The methods used are based on the theory of equivariant embeddings of spherical homogeneous spaces and reductive algebraic semigroups.