BibliographyX/ /H = Spec k[X] H , where H : X is an action of an algebraic group on an affine variety, and k[X] H is finitely generated.Other notation is gradually introduced in the text.Remark 1.2. If H ⊳ G, then G/H is equipped with the structure of a linear algebraic group with usual properties of the quotient group. Indeed, in the notation of Chevalley's theorem, we may assume that V = χ∈X(G) V χ and consider the natural linear action G : L(V ) by conjugation. The subspace E = L(V χ ) of operators preserving each V χ is G-stable, and the image of G in GL(V ) is isomorphic to G/H. See [Hum, 11.5] for details.Recall that the isotropy representation for an action G : X at x ∈ X is the natural representation G x : T x X by differentials of translations. For a quotient, the isotropy representation has a simple description:The isomorphism is given by the differential of the (separable) quotient map G → G/H. The right-hand representation of H is the quotient of the adjoint representation of H in g. Now we describe the group Autis an algebraic group acting on G/H regularly and freely. The action N(H)/H : G/H is induced by the action N(H) : G by right translations:Proof. The regularity of the action N(H)/H : G/H is a consequence of the universal property of quotients. Clearly, this action is free. Conversely, if φ ∈ Aut G (G/H), then φ(eH) = nH, and n ∈ N(H), because the φ-action preserves stabilizers. Finally, φ(gH) = gφ(eH) = gnH, ∀g ∈ G. Fibrations, bundles, and representationsThe concept of associated bundle is fundamental in topology. We consider its counterpart in algebraic geometry in a particular case.Let Z be an H-variety. Then H-acts on G × Z by h(g, z) = (gh −1 , hz).Definition 2.1. The quotient space G * H Z = (G × Z)/H equipped with the quotient topology and a structure sheaf which is the direct image of the sheaf of H-invariant regular functions is called the homogeneous fiber bundle over G/H associated with Z.The G-action on G × Z by left translations of the first factor commutes with the H-action and factors to a G-action on G * H Z. We denote by g * z the image of (g, z) in G * H Z and identify e * z with z. The embedding Let X be a G-variety, and Z ⊆ X an H-stable subvariety. By the universal property, we have a G-equivariant map µ : Gwhere ι is a closed embedding and π is a projection along a complete variety by Theorem 3.1.Example 2.2. Let N ⊆ g be the set of nilpotent elements and U = R u (B), a maximal unipotent subgroup of G. Then the map G * B u → N is proper and birational, see, e.g., [PV, 5.6]. This is a well-known Springer's resolution of singularities of N.

We study equivariant projective compactifications of reductive groups obtained by closing the image of a group in the space of operators of a projective representation. We describe the structure and the mutual position of their orbits under the action of the doubled group by left/right multiplications, the local structure in a neighborhood of a closed orbit, and obtain some conditions of normality and smoothness of a compactification. Our methods of research use the theory of equivariant embeddings of spherical homogeneous spaces and of reductive algebraic semigroups.Comment: 30 pages, AmSLaTeX. Bibliography: 36 item

No abstract

No abstract

The problem of describing adjacency on the set of orbits of a Borel subgroup B of a reductive group G acting on a spherical variety (that is, a G-variety with a finite number of S-orbits) is considered. The adjacency relation on the set of B-orbits generalizes the classical Bruhat order on the Weyl group. For a special class of homogeneous spherical varieties G/H , where H is a product of a maximal torus and the commutator subgroup of a maximal unipotent subgroup of the group G , a satisfactory description of the set of 5-orbits with adjacency relation is obtained.