# Covers of nilpotent orbits as degenerations of semisimple orbits

Let $G$ be a complex semisimple group, $P$ a parabolic subgroup, $\mathfrak{p}$ its Lie algebra, $\mathfrak{l}$, $\mathfrak{n}$ its Levi part and the nil radical, respectively. Let $\mathfrak{k}$ the center of $\mathfrak{l}$.

Consider a family of varieties $X_{k}$, $k\in\mathfrak{k}$, defined as follows (I don't know who introduced this first. I learned this from Vogan's article ; it's also studied in many other places. )

$X_k=G\times^P (\mathfrak{n}+k)$.

It's known that

• For generic $k$, $X_k$ is just the orbit $G\cdot k$, and
• the central fiber is $X_0=T^*(G/P)$.
• Let $Y_0$=Spec of the ring of functions of $X_0$ $Y_0$ is the closure of a "finite cover" of a Richardson nilpotent orbit $O=\mathrm{Ind}^{\mathfrak{g}}_{\mathfrak{l}} 0$.

The examples Vogan talks about is $G=Sp(4)$, with $P$ and $P'$ such that the corresponding Levi subgroups are $L=GL(2)$ and $L'=GL(1)\times Sp(2)$. Construct $Y_0$ and $Y'_0$ accordingly. Vogan says that $Y_0$ is the closure of the nilpotent orbit $O$ of Jordan type (2,2), and $Y'_0$ is the closure of a double cover of $O$.

My question is as follows: In these examples, both $\mathfrak{k}$ and $\mathfrak{k}'$ are one-dimensional, so denote elements in them $t\alpha$, $t\alpha'$ respectively ($t\in\mathbb{C}$; $\alpha$, $\alpha'$ the basis of $\mathfrak{k}$ and $\mathfrak{k'}$). For simplicity let's just consider the latter. Then there is an $I'_t$ of $A=S(\mathfrak{g})$ such that

$X'_{t\alpha'}=\mathrm{Spec} A/I'_t$

as long as $t\neq 0$. Now, suppose we take the "limit of $I'_t$ when $t\to 0$" in a suitable sense. (Write down the generators and let $t\to 0$, for example.) Denote the limits by $J'$ respectively. Then consider

$Z_0'=\mathrm{Spec} A/J'$.

What's the relation between $Z_0'$, $Y_0'$ and $Y_0$ above? For example, do the $G$-module structure on the ring of functions of $Z_0'$ and that of $Y_0'$ agree? (If I'm not mistaken, there's a natural filtration on the ring of functions on a coadjoint orbit, so it makes sense to talk about the $G$-module structure of the finite dimensional subspace whose degree is smaller than a given number; I don't think it can "jump" as you vary $t$...)

(Originally I also asked the following question: Vogan says, the ring of functions $Y'_0$ as $Sp(4)$ module has a five-dimensional irreducible component, while that of $Y_0$ doesn't. I'd like to see how it works. I asked a local mathematician here; he answered me that the number of times an irrep $V$ appears in the function ring of $T^*(G/P)$ is given by $\mathrm{dim} V^P$ due to a classic theorem (which I need to understand later). It's then easy to check that indeed the five-dimensional irrep only appears in the function ring of $Y_0'$, not in that of $Y_0$.)

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