Isotropy representation of flag manifolds

Alekseevsky, D. V.

Proceedings of the 17th Winter School "Geometry and Physics", GDML_Books, (1998), p. [13]-24 / Harvested from

Résumé

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 $M^{\prime }=G/K^{\prime }$ are equivalent if there exists an automorphism $\phi : G\rightarrow G$ such that $\phi (K)=K^{\prime }$ (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 ${\mathcal {K}}={\mathcal {T}}+{\mathcal {K}}^{\prime }$ where

Détail

MR MR1662721 | Zbl 0953.53033

EUDML-ID : urn:eudml:doc:219931
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