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

Kensuke Kondo, Kyo Nishiyama, Hiroyuki Ochiai, Kenji Taniguchi

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)113-119
Number of pages7
JournalKyushu Journal of Mathematics
Volume68
Issue number1
DOIs
Publication statusPublished - Jan 1 2014

Fingerprint

Closed Orbit
Flag Variety
Finite Type
Partial
Parabolic Subgroup
Orbit
Closed
Reductive Group
Algebraic Groups
Simple group
Automorphism
Fixed point
If and only if
Denote
Imply

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

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

In: Kyushu Journal of Mathematics, Vol. 68, No. 1, 01.01.2014, p. 113-119.

Research output: Contribution to journalArticle

Kondo, Kensuke ; Nishiyama, Kyo ; Ochiai, Hiroyuki ; Taniguchi, Kenji. / Closed orbits on partial flag varieties and double flag variety of finite type. In: Kyushu Journal of Mathematics. 2014 ; Vol. 68, No. 1. pp. 113-119.
@article{eacd0dcd4ef9405aa419812e009baf10,
title = "Closed orbits on partial flag varieties and double flag variety of finite type",
abstract = "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.",
author = "Kensuke Kondo and Kyo Nishiyama and Hiroyuki Ochiai and Kenji Taniguchi",
year = "2014",
month = "1",
day = "1",
doi = "10.2206/kyushujm.68.113",
language = "English",
volume = "68",
pages = "113--119",
journal = "Kyushu Journal of Mathematics",
issn = "1340-6116",
publisher = "Kyushu University, Faculty of Science",
number = "1",

}

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 -