Isotropy representation of flag manifolds

Alekseevsky, D. V.. "Isotropy representation of flag manifolds." Proceedings of the 17th Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1998. [13]-24. <http://eudml.org/doc/219931>.

@inProceedings{Alekseevsky1998,
abstract = {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},
author = {Alekseevsky, D. V.},
booktitle = {Proceedings of the 17th Winter School "Geometry and Physics"},
keywords = {Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)},
location = {Palermo},
pages = {[13]-24},
publisher = {Circolo Matematico di Palermo},
title = {Isotropy representation of flag manifolds},
url = {http://eudml.org/doc/219931},
year = {1998},
}

TY - CLSWK
AU - Alekseevsky, D. V.
TI - Isotropy representation of flag manifolds
T2 - Proceedings of the 17th Winter School "Geometry and Physics"
PY - 1998
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [13]
EP - 24
AB - 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
KW - Proceedings; Winter school; Geometry; Physics; Srní (Czech Republic)
UR - http://eudml.org/doc/219931
ER -