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Clarified what I meant by "independent of good characteristic"
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Paul Levy
Paul Levy
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Let $G$ be a simple algebraic group over the algebraically closed field $k$ of arbitrarypositive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$-orbits in ${\mathfrak g}$ and that the classification of these orbits is independent of goodthe same as over the complex numbers, as long as the characteristic of $k$ is good, i.e. odd if $G$ is not of type $A$, greater than $3$ if $G$ is of exceptional type, and greater than $5$ if $G$ is of type $E_8$. There is quite a lot of history to the subject, but a uniform proofuniform proof of the classification was given relatively recently by Premet.

My question concerns the closure ordering on nilpotent classes: where (if anywhere) is it established that the closure ordering is also independent of(in good characteristic) the same as over the complex numbers? This fact seems to be used in Varieties of nilpotent elements for simple Lie algebras I: good primes by the VIGRE group, but the reference is to Carter's book, and I can't find anywhere in the VIGRE paper where they mention a justification for why the order should be the same. (It may be in there, but I haven't found it.)

For classical Lie algebras outside characteristic 2, I think it is reasonably straightforward to use the partition type classification to show that the closure order is also independent of the characteristic. So this is really a question about exceptional Lie algebras.

Let $G$ be a simple algebraic group over the algebraically closed field $k$ of arbitrary characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$-orbits in ${\mathfrak g}$ and that the classification of these orbits is independent of good characteristic. There is quite a lot of history to the subject, but a uniform proof of the classification was given relatively recently by Premet.

My question concerns the closure ordering on nilpotent classes: where (if anywhere) is it established that the closure ordering is also independent of good characteristic? This fact seems to be used in Varieties of nilpotent elements for simple Lie algebras I: good primes by the VIGRE group, but the reference is to Carter's book, and I can't find anywhere in the VIGRE paper where they mention a justification for why the order should be the same. (It may be in there, but I haven't found it.)

For classical Lie algebras outside characteristic 2, I think it is reasonably straightforward to use the partition type classification to show that the closure order is also independent of the characteristic. So this is really a question about exceptional Lie algebras.

Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$-orbits in ${\mathfrak g}$ and that the classification of these orbits is the same as over the complex numbers, as long as the characteristic of $k$ is good, i.e. odd if $G$ is not of type $A$, greater than $3$ if $G$ is of exceptional type, and greater than $5$ if $G$ is of type $E_8$. There is quite a lot of history to the subject, but a uniform proof of the classification was given relatively recently by Premet.

My question concerns the closure ordering on nilpotent classes: where (if anywhere) is it established that the closure ordering is also (in good characteristic) the same as over the complex numbers? This fact seems to be used in Varieties of nilpotent elements for simple Lie algebras I: good primes by the VIGRE group, but the reference is to Carter's book, and I can't find anywhere in the VIGRE paper where they mention a justification for why the order should be the same. (It may be in there, but I haven't found it.)

For classical Lie algebras outside characteristic 2, I think it is reasonably straightforward to use the partition type classification to show that the closure order is also independent of the characteristic. So this is really a question about exceptional Lie algebras.

added top-level tag; also bumping for visibility
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Yemon Choi
Yemon Choi
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Paul Levy
Paul Levy
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Closure order on nilpotent orbits in exceptional Lie algebras

Let $G$ be a simple algebraic group over the algebraically closed field $k$ of arbitrary characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$-orbits in ${\mathfrak g}$ and that the classification of these orbits is independent of good characteristic. There is quite a lot of history to the subject, but a uniform proof of the classification was given relatively recently by Premet.

My question concerns the closure ordering on nilpotent classes: where (if anywhere) is it established that the closure ordering is also independent of good characteristic? This fact seems to be used in Varieties of nilpotent elements for simple Lie algebras I: good primes by the VIGRE group, but the reference is to Carter's book, and I can't find anywhere in the VIGRE paper where they mention a justification for why the order should be the same. (It may be in there, but I haven't found it.)

For classical Lie algebras outside characteristic 2, I think it is reasonably straightforward to use the partition type classification to show that the closure order is also independent of the characteristic. So this is really a question about exceptional Lie algebras.