"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 ?

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.

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

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?"

"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 ?

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.

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

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?"

"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 ?

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.

"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 ?

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.

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

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?"