"Examples first:"


Consider so(3,C). (Co)Adjoint Orbits can be described by equations 
x^2+y^2+z^2 = R.


R=0 - is nilpotent cone - algebraic closure of the orbit of nilpotent element. (It is union of two orbits - {0} and {Cone w/o {0}} ).


R$\ne$ 0 are orbits of semi-simple elements.
So we have degeneration R->0 - semi-simple orbit degenerates to nilpotent.


**Question** Is there similar description for the other nilpotent orbits in higher dimensions e.g. for gl(n,c) ? I mean can we write some equations depending on parameters F\_t(g)=0,
such that for general "t" we get semi-simple orbits, but for specific values we have nilpotent orbit (more precisely their closures)? (Here "t" can be vector and F is vector-valued algebraic function).


Of course this can be done the biggest orbit - for nilpotent cone itself.


Consider matrices "M" which satisfy the condition, that 
their characterestic polynom is fixed with values eigs $a\_i$:


$det(M-x) = (x-a\_1)(x-a\_2)...(x-a\_n)$


For $a\_i$ generic - this is semisimple orbit, but if $a\_i = 0$ we get nilpotent cone.


**Question Reformulated** Is it possible to do the same for smaller dimensional orbits ?




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As far as I heard nilpotent orbits can be described by the equations on their rank and 
$M^l=0$, however this does not seems to answer the question.




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Part of motivation for asking is related to the following questions:


[On an affine analogue of the fact $\mathbb{C}[\mathfrak{g}]^G$ is a polynomial ring](https://mathoverflow.net/questions/83926/on-an-affine-analogue-of-the-fact-mathbbc-mathfrakgg-is-a-polynomial-ri)


[Primitive ideals of the universal enveloping algebras of affine Lie algebras](https://mathoverflow.net/questions/84317/primitive-ideals-of-the-universal-enveloping-algebras-of-affine-lie-algebras)


In particular if the answer would be YES - then probably we can do the same in the
"affine case" so answering the question "What replaces the concept of the nilpotent orbit in that case?"



Are nilpotent orbits degenerations of semi-simple orbits ?
The answer is often "yes". Here is the sketch of how to obtain a nilpotent orbit as a degeneration of semisimple orbits in the $GL\_n$ case. Let $d$ be a partition of $n$ with $k$ parts and $\overline{\mathcal{O}}\_{d'}$ be the closure of the conjugacy class of nilpotent $n\times n$ matrices with Jordan blocks sizes given by the dual partition $d'.$ Denote by $\mathcal{O}\_d(t\_1,\ldots, t\_k)$ the conjugacy class of the block diagonal matrix with scalar diagonal blocks $t\_i I\_{d\_i}.$ Then 


$$\lim\_{t\to 0}\ \mathcal{O}\_d(t\_1,\ldots, t\_k)=\overline{\mathcal{O}}\_{d'}.$$


This is manifested on the level of defining equations using Oshima's approach from 



> 
> A quantization of conjugacy classes of matrices. Adv. Math. 196 (2005), no. 1, 124–146. 
> 
> 
> 


For a general $\mathfrak{g},$ this amounts to the induction of (zero-dimensional) orbits and to the correspondence between semisimple and regular orbits. In particular, every Richardson nilpotent orbit can be obtained as a degeneration in the same way. However, the defining equations are not known to the same degree of explicitness. 


On the other hand, if $\mathfrak{g}$ is simple of type other than "A" then the minimal 
nilpotent orbit is *rigid*, meaning that it cannot be deformed within the family of adjoint orbits. Existence of rigid orbits makes quantization of orbits a non-trivial task, since a very natural prescription for quantization of semisimple orbits needs to be supplemented by ad hoc quantizations of rigid orbits (several papers of Joseph addressed this question). Rigid orbits have been completely classified: if my memory serves, the answer is in Collingwood-McGovern. 


