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Algebras of curvature forms on homogeneous manifolds

By Alexander Postnikov, Boris Shapiro and Mikhail Shapiro


Abstract. Let C(X) be the algebra generated by the curvature two-forms of standard holomorphic hermitian line bundles over the complex homogeneous manifold X = G/B. The cohomology ring of X is a quotient of C(X). We calculate the Hilbert polynomial of this algebra. In particular, we show that the dimension of C(X) is equal to the number of independent subsets of roots in the corresponding root system. We also construct a more general algebra associated with a point on a Grassmannian. We calculate its Hilbert polynomial and present the algebra in terms of generators and relations. 1. Homogeneous Manifolds In this section we remind the reader the basic notions and notation related to homogeneous manifolds G/B and root systems, as well as fix our terminology. Let G be a connected complex semisimple Lie group and B its Borel subgroup. The quotient space X = G/B is then a compact homogeneous complex manifold. We choose a maximal compact subgroup K of G and denote by T = K ∩ B its maximal torus. The group K acts transitively on X. Thus X can be identified with the quotient space K/T. By g we denote the Lie algebra of G and by h ⊂ g its Cartan subalgebra. Also denote by gR ⊂ g the real form of g such that i gR is the Lie algebra of K. Analogously, hR = h ∩ gR and i hR is the Lie algebra of the maximal torus T.The root system associated with g is the set ∆ of nonzero vectors (roots) α ∈ h ∗ for which the root spaces gα = {x ∈ g | [h, x]=α(h)x for all h ∈ h} are nontrivial. Then g decomposes into the direct sum of subspaces g = h ⊕ � gα. α∈∆ For α ∈ ∆, the spaces gα and hα =[gα, g−α] are one-dimensional and there exists a unique element hα ∈ hα such that α(hα) =2. Theelementshα∈hare called coroots. Actually, α ∈ h ∗ R and hα ∈ hR, forα∈∆. Let us choose generators eα ∈ gR of the root spaces gα such that [eα,e−α] =hα for any root α. Then[hα,eα] =2eα and [hα,e−α] =−2e−α. The root system ∆ is subdivided into a disjoint union of sets of positive roots ∆+ and negative roots ∆ − = −∆+ such that the direct sum b = h ⊕ � α∈∆+ gα is th

Topics: reflections sα, h ∗ → h ∗, α ∈ ∆, givenby
Year: 1999
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