By Jules Molk (ed)
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Additional info for Encyclopédie des sciences mathématiques, tome IV-volume 2
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.
Encyclopédie des sciences mathématiques, tome IV-volume 2 by Jules Molk (ed)