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Isotropy representation of flag manifolds. (English) Zbl 0953.53033
Slovák, Jan (ed.) et al., Proceedings of the 17th winter school “Geometry and physics”, Srní, Czech Republic, January 11-18, 1997. Palermo: Circolo Matematico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 54, 13-24 (1998).
A flag manifold of a compact semisimple Lie group $$G$$ is defined as a quotient $$M=G/K$$ where $$K$$ is the centralizer of a one-parameter subgroup $$\exp(tx)$$ of $$G$$. Then $$M$$ can be identified with the adjoint orbit of $$x$$ in the Lie algebra $$\mathcal G$$ of $$G$$. Two flag manifolds $$M=G/K$$ and $$M'=G/K'$$ are equivalent if there exists an automorphism $$\phi: G\to G$$ such that $$\phi(K)=K'$$ (equivalent manifolds need not be $$G$$-diffeomorphic since $$\phi$$ is not assumed to be inner). In this article, explicit formulas for decompositions of the isotropy representation for all flag manifolds appearing in algebras of the classical series $$A$$, $$B$$, $$C$$, $$D$$, are derived. The answer involves painted Dynkin graphs which, by a result of the author [“Flag manifolds”, Reprint ESI 415, (1997) see also Zb. Rad., Beogr. 6(14), 3–35 (1997; Zbl 0946.53025)], classify flag manifolds. The Lie algebra $$\mathcal K$$ of $$K$$ admits the natural decomposition $${\mathcal K}={\mathcal T}+{\mathcal K}'$$ where $${\mathcal T}$$ is the center of $${\mathcal K}$$ and $${\mathcal K}'=[{\mathcal K},{\mathcal K}]$$. Let $$R$$ be the root system of $$\mathcal G$$, $$R_K$$ be the set of roots orthogonal to $$x$$ (the subsystem of $$R$$ corresponding to $$\mathcal K'$$), and $$R_M=R\setminus R_K$$. Each set of simple roots in $$R_K$$ admits an extension (not unique in general) to the base of $$R$$. This defines the painting. It reduces to some standard form which is explicitely described. The restriction of a root in $$R_M$$ to $$\mathcal T$$ is called a $$T$$-root. A root $$\alpha\in R_M$$ is called $$K$$-simple if $$\alpha-\beta\notin R$$ for any $$\beta\in R_K^+$$ where $$R_K^+$$ is the set of positive roots. It is known [J. de Siebenthal, Comment. Math. Helv. 44, 1–44 (1969; Zbl 0185.08601), D. V. Alekseevsky and A. M. Perelomov, Funct. Anal. Appl. 20, 171–182 (1986; Zbl 0641.53050)] that irreducible $$K$$-submodules of the complexified tangent space are parametrized by $$T$$-roots or, equivalently, $$K$$-simple roots, that submodules corresponding to opposite roots are conjugated, and that the real form of the sum of the latter two modules is irreducible. The decomposition is given in these terms by the direct calculation of $$K$$-simple roots for standard painted Dynkin graphs of the classical series.
For the entire collection see [Zbl 0904.00040].
Reviewer: V.Gichev (Omsk)

MSC:
 53C30 Differential geometry of homogeneous manifolds 22E46 Semisimple Lie groups and their representations