### Abstract

Let G be a connected reductive algebraic group over C. We denote by K = (G^{θ})_{0}the identity component of the fixed points of an involutive automorphism θ of G. The pair (G, K) is called a symmetric pair. Let Q be a parabolic subgroup of K. We want to find a pair of parabolic subgroups P _{1}, P _{2}of G such that (i) P _{1}∩ P _{2}= Q and (ii) P _{1} P _{2}is dense in G. The main result of this article states that, for a simple group G, we can find such a pair if and only if (G, K) is a Hermitian symmetric pair. The conditions (i) and (ii) imply that the K -orbit through the origin (eP _{1}, eP _{2}) of G/P _{1}× G/P _{2}is closed and it generates an open dense G -orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed K -orbits on G/P _{1}× G/P _{2}.

Original language | English |
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Pages (from-to) | 113-119 |

Number of pages | 7 |

Journal | Kyushu Journal of Mathematics |

Volume | 68 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 1 2014 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Kyushu Journal of Mathematics*,

*68*(1), 113-119. https://doi.org/10.2206/kyushujm.68.113

**Closed orbits on partial flag varieties and double flag variety of finite type.** / Kondo, Kensuke; Nishiyama, Kyo; Ochiai, Hiroyuki; Taniguchi, Kenji.

Research output: Contribution to journal › Article

*Kyushu Journal of Mathematics*, vol. 68, no. 1, pp. 113-119. https://doi.org/10.2206/kyushujm.68.113

}

TY - JOUR

T1 - Closed orbits on partial flag varieties and double flag variety of finite type

AU - Kondo, Kensuke

AU - Nishiyama, Kyo

AU - Ochiai, Hiroyuki

AU - Taniguchi, Kenji

PY - 2014/1/1

Y1 - 2014/1/1

N2 - Let G be a connected reductive algebraic group over C. We denote by K = (Gθ)0the identity component of the fixed points of an involutive automorphism θ of G. The pair (G, K) is called a symmetric pair. Let Q be a parabolic subgroup of K. We want to find a pair of parabolic subgroups P 1, P 2of G such that (i) P 1∩ P 2= Q and (ii) P 1 P 2is dense in G. The main result of this article states that, for a simple group G, we can find such a pair if and only if (G, K) is a Hermitian symmetric pair. The conditions (i) and (ii) imply that the K -orbit through the origin (eP 1, eP 2) of G/P 1× G/P 2is closed and it generates an open dense G -orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed K -orbits on G/P 1× G/P 2.

AB - Let G be a connected reductive algebraic group over C. We denote by K = (Gθ)0the identity component of the fixed points of an involutive automorphism θ of G. The pair (G, K) is called a symmetric pair. Let Q be a parabolic subgroup of K. We want to find a pair of parabolic subgroups P 1, P 2of G such that (i) P 1∩ P 2= Q and (ii) P 1 P 2is dense in G. The main result of this article states that, for a simple group G, we can find such a pair if and only if (G, K) is a Hermitian symmetric pair. The conditions (i) and (ii) imply that the K -orbit through the origin (eP 1, eP 2) of G/P 1× G/P 2is closed and it generates an open dense G -orbit on the product of partial flag variety. From this point of view, we also give a complete classification of closed K -orbits on G/P 1× G/P 2.

UR - http://www.scopus.com/inward/record.url?scp=84887624864&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84887624864&partnerID=8YFLogxK

U2 - 10.2206/kyushujm.68.113

DO - 10.2206/kyushujm.68.113

M3 - Article

VL - 68

SP - 113

EP - 119

JO - Kyushu Journal of Mathematics

JF - Kyushu Journal of Mathematics

SN - 1340-6116

IS - 1

ER -