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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 $\cal G$ 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 $\cal K$ of $K$ admits the natural decomposition ${\cal K}={\cal T}+{\cal K}'$ where (English) |
