By D.A. Timashev

Homogeneous areas of linear algebraic teams lie on the crossroads of algebraic geometry, thought of algebraic teams, classical projective and enumerative geometry, harmonic research, and illustration concept. by way of average purposes of algebraic geometry, with the intention to clear up a variety of difficulties on a homogeneous house, it really is normal and beneficial to compactify it whereas keeping an eye on the gang motion, i.e., to contemplate equivariant completions or, extra often, open embeddings of a given homogeneous house. Such equivariant embeddings are the topic of this ebook. We specialise in the class of equivariant embeddings by way of convinced information of "combinatorial" nature (the Luna-Vust thought) and outline of assorted geometric and representation-theoretic homes of those kinds in line with those facts. the category of round types, intensively studied over the past 3 a long time, is of precise curiosity within the scope of this publication. round forms contain many classical examples, similar to Grassmannians, flag types, and forms of quadrics, in addition to famous toric kinds. we've got tried to hide lots of the very important concerns, together with the new vast development acquired in and round the idea of round varieties.

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Fr ∈ k(X) be rational T -eigenfunctions whose weights generate Λ(X). 1, Λ(X) = Λ(T x). 1. The function x → r(Gx) is lower semicontinuous on X. 4 ([Arzh, §2]). The function x → c(Gx) is lower semicontinuous on X. CHAPTER 2. COMPLEXITY AND RANK 30 In the aﬃne case, the weight semigroup is a more subtle invariant of an action than the weight lattice. 5. For quasiaffine X, Λ(X) = ZΛ+ (X). Proof. Any rational B-eigenfunction on X is a quotient of two polynomials: f = f1 /f2 . By the Lie-Kolchin theorem, there exists a nonzero Bsemiinvariant linear combination λi (bi f2 ), λi ∈ k, bi ∈ B.

It is easy to see that complexities of G/H, G/HZ, and G′ /(HZ ∩ G′ ) are equal. Therefore it suﬃces to solve the classiﬁcation problem for semisimple G. 1) dim H ≤ dim U − c A more subtle restriction is based on the notion of d-decomposition [Pan2]. A triple of reductive groups (L, L1 , L2 ) is called a d-decomposition if dL1 ×L2 (L) = d, where L1 × L2 acts on L by left and right multiplications. Clearly, CHAPTER 2. COMPLEXITY AND RANK 53 (L, L1 , L2 ) remains a d-decomposition if one permutes L1 , L2 or replaces them by conjugates.

It follows that a ≃ a∗ intersects all closed G-orbits in MG/S , and LG/S = πG (a∗ ), where πG : g∗ → g∗ //G is the quotient map. Finally, generic ﬁbers of Φ are ﬁnite. Indeed, it suﬃces to ﬁnd at least −1 − − one ﬁnite ﬁber. 1]. 1. For any horospherical G-variety X of type S, the natural map G ∗P − s⊥ → MX = Gs⊥ is generically finite, proper and surjective, and LX = πG (a∗ ). We have already seen that horospherical varieties, their cotangent bundles and moment maps are easily accessible for study.

### Homogeneous Spaces and Equivariant Embeddings by D.A. Timashev

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