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Flag manifolds. (English) Zbl 0946.53025

Bokan, Neda (ed.) et al., 11th Yugoslav geometrical seminar, Divčibare, Yugoslavia, October 10-17, 1996. Invited papers. Beograd: Matematički Institut SANU. Zb. Rad., Beogr. 6(14), 3-35 (1997).
This survey article grew out of a mini course given at the 11th Yugoslav Geometrical Seminar, Fall School of Differential geometry, Divčibare 1996.
In the first part, the standard review of Lie groups and Lie algebras is given (definitions, structure of Lie groups and Lie algebras, and structure of complex semisimple Lie algebras). In the second part, homogeneous manifolds \(M=G/K\) are studied, in particular the integrability of an invariant almost complex structure \(J\) and almost symplectic structure \(\omega\).
In the third part, examples of flag manifolds are given in more detail. Firstly, manifolds of complex flags and complex isotropic flags are described as coset spaces of the classical Lie groups, \(SL_n({\mathbb C})\), \(SO_n({\mathbb C})\), \(Sp_n({\mathbb C})\), modulo parabolic subgroups. A subgroup \(P\) of \(G\) is called parabolic if it contains a Borel subgroup (i.e., a maximal solvable subgroup) \(B\) of \(G\). A flag manifold of a complex semisimple Lie group is the quotient \(M=G/P\) of \(G\) by a parabolic subgroup \(P\). The action of a maximal compact subgroup of a flag manifold is studied.
In the last, fourth part, flag manifolds are studied as quotients of compact semisimple Lie groups. Using the notions of \(T\)-roots, \(T\)-chambers and \(T\)-Weyl groups of a flag manifold \(M=G^\tau/P^\tau\), a realization of a flag manifold in terms of an adjoint orbit is given. Here, \(\tau\) denotes the standard antiholomorphic involution of \(G\) which determines a compact form \(G^\tau\) of \(G\), and \(M=G/P=G^\tau/P^\tau\). It is known that there exists a one-to-one correspondence between \(T\)-chambers and invariant complex structures on a flag manifold \(G^\tau/K^\tau\) (defined up to sign) [D. V. Alekseevskij and A. M. Perelomov, Funct. Anal. Appl. 20, No. 3, 171-182 (1986; Zbl 0641.53050)]. The notion of Koszul numbers is important for the understanding of invariant Kähler-Einstein metrics. For the classical Lie algebras \(A_l\), \(B_l\), \(C_l\), \(D_l\), the Koszul numbers are determined by Alekseevsky and Perelemov [ibid.]. This is important since the Koszul numbers are invariants of an invariant complexe structure, i.e., they can be used to determine the number of not diffeomorphic invariant complex structures on a flag manifold.
For the entire collection see [Zbl 0870.00034].

MSC:

53C35 Differential geometry of symmetric spaces
53C55 Global differential geometry of Hermitian and Kählerian manifolds
53C56 Other complex differential geometry

Citations:

Zbl 0641.53050
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