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How would one classify the strata for the standard nilpotent cone for $GL_{k}(\mathbb{C})$, using the definition from Hesselink's paper "Desingularizations of Varieties of Nullforms"Nullforms"? I know that they correspond to partitions / nilpotent orbits etc, but from first principles why aren't two different nilpotent orbits possibly in the same strata - how would you prove that? (preferably using the definition of Hesselink)

I would like to classify the strata for the problem I'm working on, but don't completely understand how to do it for the more basic case (for which the method is probably well-known), I get stuck on the details, so that would be very helpful. Thanks.

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# Classifying strata for the adjoint representation of GL from first principles

How would one classify the strata for the standard nilpotent cone for $GL_{k}(\mathbb{C})$, using the definition from Hesselink's paper "Desingularizations of Varieties of Nullforms"? I know that they correspond to partitions / nilpotent orbits etc, but from first principles why aren't two different nilpotent orbits possibly in the same strata - how would you prove that? (preferably using the definition of Hesselink)

I would like to classify the strata for the problem I'm working on, but don't completely understand how to do it for the more basic case (for which the method is probably well-known), I get stuck on the details, so that would be very helpful. Thanks.