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A question about centralizer of a vectors in the positive Weyl chamber
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Given a compact Lie group $K$ and a maximal torus $Tleq K$, and choose a positive Weyl chamber $mathfrak t^*_+subsetmathfrak k^*$, where we used a $K$-invariant inner product on $mathfrak k$. Then $mathfrak t^*_+$ can be decomposed into disjoint open faces
beginequation
mathfrak t^*_+=bigsqcup_sigmainSigmasigma
endequation
where $Sigma$ is the set of all faces.
Then question is about centralizer $K_xi$ of $xiinsigmainSigma$, do we have $K_xi$ only depends on $sigma$ and not depends on $xiinsigma$?
lie-groups lie-algebras
asked Aug 9 at 8:11
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1211
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up vote
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Given a compact Lie group $K$ and a maximal torus $Tleq K$, and choose a positive Weyl chamber $mathfrak t^*_+subsetmathfrak k^*$, where we used a $K$-invariant inner product on $mathfrak k$. Then $mathfrak t^*_+$ can be decomposed into disjoint open faces
beginequation
mathfrak t^*_+=bigsqcup_sigmainSigmasigma
endequation
where $Sigma$ is the set of all faces.
Then question is about centralizer $K_xi$ of $xiinsigmainSigma$, do we have $K_xi$ only depends on $sigma$ and not depends on $xiinsigma$?
lie-groups lie-algebras
asked Aug 9 at 8:11
Display Name
1211
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Given a compact Lie group $K$ and a maximal torus $Tleq K$, and choose a positive Weyl chamber $mathfrak t^*_+subsetmathfrak k^*$, where we used a $K$-invariant inner product on $mathfrak k$. Then $mathfrak t^*_+$ can be decomposed into disjoint open faces
beginequation
mathfrak t^*_+=bigsqcup_sigmainSigmasigma
endequation
where $Sigma$ is the set of all faces.
Then question is about centralizer $K_xi$ of $xiinsigmainSigma$, do we have $K_xi$ only depends on $sigma$ and not depends on $xiinsigma$?
lie-groups lie-algebras
asked Aug 9 at 8:11
Display Name
1211
Given a compact Lie group $K$ and a maximal torus $Tleq K$, and choose a positive Weyl chamber $mathfrak t^*_+subsetmathfrak k^*$, where we used a $K$-invariant inner product on $mathfrak k$. Then $mathfrak t^*_+$ can be decomposed into disjoint open faces
beginequation
mathfrak t^*_+=bigsqcup_sigmainSigmasigma
endequation
where $Sigma$ is the set of all faces.
Then question is about centralizer $K_xi$ of $xiinsigmainSigma$, do we have $K_xi$ only depends on $sigma$ and not depends on $xiinsigma$?
lie-groups lie-algebras
asked Aug 9 at 8:11
Display Name
1211
asked Aug 9 at 8:11
Display Name
1211
asked Aug 9 at 8:11
Display Name
1211
asked Aug 9 at 8:11
Display Name
1211
1211
add a comment |Â
add a comment |Â
1 Answer
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The identity component of the centralizer depends only on the face, but the component group is more sensitive. For proof of these statements, an example of the dependence, and an explanation of how to compute both, I refer to §2.4 of Mark Reeder's excellent survey, "Torsion automorphisms of simple Lie algebras".
answered Aug 9 at 8:57
Gro-Tsen
8,96323283
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
The identity component of the centralizer depends only on the face, but the component group is more sensitive. For proof of these statements, an example of the dependence, and an explanation of how to compute both, I refer to §2.4 of Mark Reeder's excellent survey, "Torsion automorphisms of simple Lie algebras".
answered Aug 9 at 8:57
Gro-Tsen
8,96323283
add a comment |Â
up vote
4
down vote
The identity component of the centralizer depends only on the face, but the component group is more sensitive. For proof of these statements, an example of the dependence, and an explanation of how to compute both, I refer to §2.4 of Mark Reeder's excellent survey, "Torsion automorphisms of simple Lie algebras".
answered Aug 9 at 8:57
Gro-Tsen
8,96323283
add a comment |Â
up vote
4
down vote
up vote
4
down vote
The identity component of the centralizer depends only on the face, but the component group is more sensitive. For proof of these statements, an example of the dependence, and an explanation of how to compute both, I refer to §2.4 of Mark Reeder's excellent survey, "Torsion automorphisms of simple Lie algebras".
answered Aug 9 at 8:57
Gro-Tsen
8,96323283
The identity component of the centralizer depends only on the face, but the component group is more sensitive. For proof of these statements, an example of the dependence, and an explanation of how to compute both, I refer to §2.4 of Mark Reeder's excellent survey, "Torsion automorphisms of simple Lie algebras".
answered Aug 9 at 8:57
Gro-Tsen
8,96323283
answered Aug 9 at 8:57
Gro-Tsen
8,96323283
answered Aug 9 at 8:57
Gro-Tsen
8,96323283
answered Aug 9 at 8:57
Gro-Tsen
8,96323283
8,96323283
add a comment |Â
add a comment |Â
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