# 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 ...

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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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### 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&...

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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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### Two-sided cells, special nilpotent orbits and special representations

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. This question concerns three classical objects of representation theory: the two-sided Kazhdan-Lusztig cells of the Weyl group $W$ of $\mathfrak{...

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### Regular nilpotents and minimal parabolic subalgebras in real semisimple Lie algebras

Let $\mathfrak{g}$ be a real semisimple Lie algebra. A subalgebra $\mathfrak{p}$ of $\mathfrak{g}$ is parabolic if its complexification is parabolic in $\mathfrak{g}_\mathbb{C}$, meaning it contains a ...

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### Slodowy slice intersecting a given orbit "minimally"?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Is it true that for any $X\in\mathfrak{g}$, there exists an $\mathfrak{sl}_2$-triple $(e,h,f)$ in $\mathfrak{g}$ such that
We have $X\in e+Z_{\...

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### Normalizers over centralizers of Levi subalgebras

Let $\mathfrak g$ be a simple Lie algebra and $e$ a nilpotent element of $\mathfrak g$. Using Jacobson-Morozov theorem, there exist a nilpotent element $f$ and a semisimple element $h$ such that $\{e,...

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### Nahm's equations with poles and conservation of characteristic polynomial

In "Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory" Nahm's equations are studied in Section 3. In particular, it is explained that their moduli space of solutions, $\...

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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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### Classification of real nilpotent orbits using real parabolic subalgebras?

Bala and Carter's approach to the classification of nilpotent orbits in complex semisimple Lie algebras $\mathfrak g$ is to associate to each nilpotent element $e\in\mathfrak g$ the pair $(\mathfrak l,...

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### Vanishing of power of nilpotent operator $\mathrm{ad} \, \;e$ in different characteritics

This question needs some background:
(1) In his influential 1959 paper here, Kostant studied the adjoint representation of a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ (which can be ...

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### Affine analog of the theory of sheets

In the study of adjoint orbits in a complex semi-simple lie algebra, there is a well known object known as a "sheet". These are the irreducible components of the union of orbits of the same dimension. ...

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### Differential of the adjoint quotient map

My question is regarding a paper by R.W Richardson titled "Derivatives of invariant polynomials on a semisimple Lie Algebra" ** . In this paper, he reports on computations of the rank of the ...

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### Equality of codimension under Lusztig-Spaltenstein induction

Pardon if this is well known. Suppose I have a (say complex) connected reductive group $G^{\vee}$ with the $\tilde{\Delta}=\Delta\cup\{\alpha_0\}$ being the simple roots plus the negative highest root ...

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### Wavefront sets of irreducible representations with non-integral infinitesimal characters

Let $G$ be a complex reductive algebraic group (connected, simply connected etc), viewed as a real group. We study the representations of $G$, and we follow the notations in the paper of Barbasch and ...

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### Closure order on nilpotent orbits in exceptional Lie algebras

Let $G$ be a simple algebraic group over the algebraically closed field $k$ of positive characteristic, and let ${\mathfrak g}={\rm Lie}(G)$. It is well known that there are finitely many nilpotent $G$...

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### Conjugacy classes of a triple

In the paper " THE COMPONENT GROUPS OF NILPOTENTS IN EXCEPTIONAL SIMPLE REAL LIE ALGEBRAS" by D. King I am unable to proof the lemma 3.7 which is omitted there.
The lemma is following:
Let $\...

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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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### the affine coordinate ring of orbit closures in the ordinary nilpotent cone

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 ...