Homogeneous spaces of linear algebraic groups lie at the
crossroads of algebraic geometry, theory of algebraic groups,
classical projective and enumerative geometry, harmonic analysis,
and representation theory. By standard reasons of algebraic
geometry, in order to solve various problems on a homogeneous
space, it is natural and helpful to compactify it while keeping
track of the group action, i.e., to consider equivariant
completions or, more generally, open embeddings of a given
homogeneous space. Such equivariant embeddings are the subject of
this book. We focus on the classification of equivariant embeddings
in terms of certain data of "combinatorial" nature (the Luna-Vust
theory) and description of various geometric and
representation-theoretic properties of these varieties based on
these data. The class of spherical varieties, intensively studied
during the last three decades, is of special interest in the scope
of this book. Spherical varieties include many classical examples,
such as Grassmannians, flag varieties, and varieties of quadrics,
as well as well-known toric varieties. We have attempted to cover
most of the important issues, including the recent substantial
progress obtained in and around the theory of spherical
varieties.
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