|
8
|
|
|
Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional
and each of its fibres is an trreducible complete intersection and contains a unique open
$G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as
$G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).
However this example does not answer the question as the stabilisers $G_x$ of regular elements $x\in g$ are not self-normalising (I have completely overlooked the extra condition on $x$ in the first reading, which implies that $N_G(G_x)=G_x$).
One could try the following approach instead:
Take two non-conjugate semisimple maximal subgroups $H_1$ and $H_2$ in $G$ (there are quite a few of them, if $G$ is big, and they were classified by Dynkin in the 50s). Then consider the disjoint union of homogeneous spaces $X=(G/H_1)\sqcup (G/H_2)$. The set $X$ has a natural structure of a an affine $G$-variety (with two irreducible components) and for any point of $x\in X$ we have $X^{G_x}=${$x$}. If we take $H_1$ small (say such that $H_1^\circ\cong SL_2$)
and $H_2$ very big, then the multiplicities for $H_1$ would be big and for $H_2$ they would be very small (maybe even $\le 1$ if we choose $G$ and $H_2$ carefully).
|
|
|
|
|
7
|
|
|
Here is a counterexample: Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional However this example does not answer the question as the stabilisers $G_x$ of regular elements $x\in g$ are not self-normalising (I have completely overlooked the extra condition on $x$ in the first reading, which implies that $N_G(G_x)=G_x$). One can do similar things with all nonzero semisimple orbitscould try the following approach instead: they Take two non-conjugate semisimple maximal subgroups $H_1$ and $H_2$ in $G$ (there are closed but certain one parameter families quite a few of such orbits always degenerate to nilpotent ones (called Richardson)them, if $G$ is big, and they were classified by Dynkin in the 50s). All multiplicities will be Then consider the same as disjoint union of homogeneous spaces $X=(G/H_1)\sqcup (G/H_2)$. The set $X$ has a natural structure of a an affine $G$-variety (with two irreducible components) and for those Richardson orbits provided that the latter any point of $x\in X$ we have a normal closure $X^{G_x}=${$x$}. If we take $H_1$ small (which is always say such that $H_1^\circ\cong SL_2$)and $H_2$ very big, then the case multiplicities for $g=sl_n$). The original elements being semisimple there will H_1$ would be no inclusion to big and for $\mathcal N$, of course. More on this can H_2$ they would be found in Section 8 of Jantzen's notes on nilpotent orbitsvery small (maybe even $\le 1$ if we choose $G$ and $H_2$ carefully).
|
|
|
|
|
6
|
|
|
Here is a counterexample:
Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional
and each of its fibres is an trreducible complete intersection and contains a unique open
$G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as
$G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).
One can do similar things with all nonzero semisimple orbits: they are closed but certain one parameter families of such orbits always degenerate to nilpotent ones (called Richardson). All multiplicities will be the same as for those Richardson orbits provided that the latter have a normal closure (which is always the case for $g=sl_n$).
The originsl original elements being semisimple there will be no inclusion to $\mathcal N$, of course. More on this can be found in Section 8 of Jantzen's notes on nilpotent orbits.
|
|
|
|
5
|
|
|
Here is a counterexample:
Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional
and each of its fibres is an trreducible complete intersections intersection and contains a unique open
$G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as
$G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).
One can do similar things with all nonzero semisimple orbits: they are closed but their certain one parameter families of such orbits always degenerate to nilpotent ones (called Richardson). All multiplicities will be the same as for those Richardson orbits provided that the latter have a normal closure (which is always the case for $g=sl_n$).
The originsl elements being semisimple there will be no inclusion to $\mathcal N$, of course. More on this can be found in Section 8 of Jantzen's notes on nilpotent orbits.
|
|
|
|
|
4
|
|
|
Here is a counterexample:
Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional
and each of its fibres is an trreducible complete intersections and contains a unique open
$G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as
$G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).
One can do similar things with all nonzero semisimple orbits: they are closed and deform but their one parameter families always degenerate to nilpotent ones (called Richardson). The All multiplicities will be the same as for those nilpotent Richardson orbits provided that the latter have a normal closure (which is always the case for $g=sl_n$).
The originsl elements being semisimple there will be no inclusion to $\mathcal N$, of course. More on this can be found in Section 8 of Jantzen's notes on nilpotent orbits.
|
|
|
|
|
3
|
|
|
Here is a counterexample:
Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional
and each of its fibres is an trreducible complete intersections and contains a unique open
$G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as
$G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).
One can do similar things with all nonzero semisimple orbits: they are closed and deform to nilpotent ones (called Richardson). The multiplicities will be the same as for those nilpotent orbits , but provided that the latter have a normal closure (which is always the case for $g=sl_n$).
The elements being semisimple ther there will be no inclusion to $\mathcal N$, of course. More on this can be found in Section 8 of Jantzen's notes on nilpotent orbits.
|
|
|
|
|
2
|
|
|
Here is a counterexample:
Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional
and each of its fibres is an trreducible complete intersections and contains a unique open
$G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as
$G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).
One can do similar things with all nonzero semisimple orbits: they are closed and deform to nilpotent ones (called Richardson). The multiplicities will be the same as for those nilpotent orbits, but the elements being semisimple ther will be no inclusion to $\mathcal N$.
|
|
|
|
|
1
|
|
|
Here is a counterexample:
Take the nilpotent cone $\mathcal N$ in $g={\rm Lie}(G)$ and the $G$-orbit of a regular semisimple element, $h$ say. The categorical quotient $g\rightarrow g//G\cong\mathbb{A}^l$, $l={\rm rk}(G),$ is equidimensional
and each of its fibres is an trreducible complete intersections and contains a unique open
$G$-orbit. Since the orbit $Gh$ is closed it coincides with one of the fibres and the algebra of regular functions $\mathbb{C}[Gh]$ is just a filtered deformation of the graded algebra $\mathbb{C}[\mathcal{N}]=\mathbb{C}[g]/(f_1,\ldots,f_l)$ whose defining ideal is generated by $f_1-\lambda_1,\ldots, f_l-\lambda_l$ for some $\lambda_i\in\mathbb{C}$ (here $f_1,\ldots, f_l$is a set of free homogeneous generators for $\mathbb{C}[g]^G$). Since $G$ is reductive, we are in characteristic $0$ and the action of $G$ on $\mathbb{C}[Gh]$ is rational, we have that $\mathbb{C}[\mathcal{N}]\cong \mathbb{C}[Gh]$ as
$G$-modules. So all multiplicities will be the same in both cases. However, $Gh$ is not contained in $\mathcal N$ (and vice versa).
One can do similar things with all semisimple orbits: they are closed and deform to nilpotent ones (called Richardson). The multiplicities will be the same as for those nilpotent orbits, but the elements being semisimple ther will be no inclusion to $\mathcal N$.
|
|
|
