# Questions tagged [nilpotent-orbits]

These are the adjoint orbits of a complex semisimple group lying in the nilpotent cone. Nilpotent orbits arise in algebraic geometry, symplectic geometry, and representation theory.

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### Why do most eigenspaces of a Lie algebra automorphism have finitely many orbits?

I'm interested in understanding the following lemma, which Vogan states (Lemma 4.8) in his paper on the Local Langlands Conjectures (omitting the "well-known" proof). Suppose $G$ is a ...
1 vote
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### Minimal $K$-orbit on $\mathfrak{g}$

Let $\mathfrak{g}_0$ be a noncompact simple Lie algebra with Cartan decomposition $\mathfrak{g}_0=\mathfrak{k}_0+\mathfrak{p}_0$. Write $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the ...
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### Nilpotent orbits and mixed Hodge structures

Let $(H_\mathbb{Z}, \{h^{p,q}\}_{p+q=n}, \phi)$ be the datum to define weight $n$ polarized Hodge structures on $H_\mathbb{Z}$, $h^{p,q}$ is the Hodge numbers, $\phi$ is a polarization. Let $D$ be the ...
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### Transversality of Slodowy-type slice

Let $x$ be a nilpotent element in $\mathfrak{g}=\mathfrak{gl}_n$ and $y$ some arbitrary element in $\mathfrak{g}$. Let $Z_\mathfrak{g}(x)$ be the centralizer of $x$ in $\mathfrak{g}$ and consider the ...
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1 vote
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### The identity connected component of centralizers of unipotent orbits

This is, in a way, a follow up question to Unipotent orbits and intersection with Levi and pseudo-Levi subgroups. I was reading "A generalisation of the Bala–Carter theorem for nilpotent orbits&...
1 vote
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### Unipotent orbits and intersection with Levi and pseudo-Levi subgroups

Given a simple complex Lie group $G$ (I might say upfront that I am mostly interested with exceptional Lie algebras) and a nilpotent orbit $\mathcal{O}\subset G$ I would like to describe the ...
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### $\mathbb{F}_q$-rational elements in unipotent classes of a finite group of Lie-type

I've tried posting this question on MSE, but didn't manage to get an answer there, so I'm trying again here. Sorry in advance if this question is trivial or trivially false. I haven't managed to find ...
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### Kazhdan Lusztig map and Richardson orbits

Let $g$ be a simple Lie algebra with Weyl group $W$. Kazhdan and Lusztig defined a map $\Phi$: nilpotent orbits in $g$ $\rightarrow$ conjugacy classes in $W$. Let $\eta_p$ be a Richardson orbit ...
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### Number of Richardson orbits in simple Lie algebras of types $E_n$?

This is a follow-up to my question about nilpotent orbits here asked in connection with an earlier discussion of symplectic resolutions. Leaving aside the connections with algebraic geometry and ...
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### Examples of Richardson orbit closures not having a symplectic resolution?

This is a follow-up to a recent question asked by Peter Crooks here. The answer by Ben Webster includes a helpful link to the corrected arXiv version of Baohua Fu's 2003 Invent. Math. paper ...
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### Which nilpotent orbit closures admit Springer resolutions?

Let $G$ be a connected, simply-connected complex semisimple group. We have the famous Springer resolution $$T^*(G/B)\rightarrow\mathcal{N}$$ of the closure of the regular nilpotent orbit. My ...
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### Shalika germ for local function field

I am wondering if there is a theorem of Shalika germ (as below) for local function field, for both the group version or the Lie algebra version, probably under assumption on the characteristic to be ...
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### A natural bijection between the orbit spaces of $p$-nilpotent matrices for varying $p$

Let $k$ be an algebraically closed field of characteristic $p$, call a matrix $X\in\mathfrak{gl}_n(k)$ $p$-nilpotent if $X^p=0$, and let $\mathcal{N}_1=\mathcal{N}_1(\mathfrak{gl}_n(k))$ be the set of ...
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### Equivariant Stratifications of a Variety

Let $X$ be a complex variety acted upon algebraically by a complex torus $T$. Suppose that $\{X_{\beta}\}_{\beta\in S}$ is a finite $T$-equivariant stratification of $X$, so that the $X_{\beta}$ are ...
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Given a partition $\lambda$ of $n$, consider the orbit closure $\overline{ \mathcal{O}_{\lambda}}$ of the nilpotent orbit corresponding to that partition. My question, is how to explicitly construct ...