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 $exp\left(tx\right)$ of $G$. Then $M$ can be identified with the adjoint orbit of $x$ in the Lie algebra $𝒢$ of $G$. Two flag manifolds $M=G/K$ and $M^{\text{'}}=G/K^{\text{'}}$ are equivalent if there exists an automorphism $\phi :G\to G$ such that $\phi \left(K\right)=K^{\text{'}}$ (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 $𝒦$ of $K$ admits the natural decomposition $𝒦=𝒯+{𝒦}^{\text{'}}$ where

EUDML-ID : urn:eudml:doc:219931
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@article{702137,
     title = {Isotropy representation of flag manifolds},
     booktitle = {Proceedings of the 17th Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1998},
     pages = {[13]-24},
     mrnumber = {MR1662721},
     zbl = {0953.53033},
     url = {http://dml.mathdoc.fr/item/702137}
}

Alekseevsky, D. V. Isotropy representation of flag manifolds, dans Proceedings of the 17th Winter School "Geometry and Physics", GDML_Books,  (1998), pp. [13]-24. http://gdmltest.u-ga.fr/item/702137/