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.
28
questions
3
votes
0
answers
54
views
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 ...
- 477
1
vote
0
answers
66
views
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 ...
- 759
2
votes
0
answers
106
views
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 ...
- 21
3
votes
0
answers
170
views
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 ...
- 390
1
vote
1
answer
151
views
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&...
- 141
1
vote
1
answer
213
views
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 ...
- 141
11
votes
0
answers
450
views
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{...
- 251
4
votes
1
answer
177
views
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 ...
9
votes
2
answers
450
views
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_{\...
- 1,754
1
vote
0
answers
90
views
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,...
- 11
2
votes
2
answers
209
views
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, $\...
- 1,105
6
votes
0
answers
149
views
$\mathbb{F}_q$-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 ...
- 993
6
votes
1
answer
360
views
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 ...
- 2,519
6
votes
0
answers
234
views
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,...
- 4,656
4
votes
1
answer
230
views
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 ...
- 52k
4
votes
1
answer
343
views
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. ...
- 1,063
2
votes
0
answers
316
views
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 ...
- 1,063
7
votes
1
answer
359
views
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 ...
- 1,754
6
votes
1
answer
327
views
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 ...
- 73
8
votes
1
answer
572
views
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 $G$...
- 1,184
2
votes
0
answers
53
views
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 $\...
- 116
7
votes
1
answer
501
views
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 ...
- 52k
9
votes
2
answers
1k
views
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 ...
- 52k
8
votes
1
answer
644
views
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 ...
- 4,850
3
votes
0
answers
215
views
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 ...
- 1,754
2
votes
2
answers
335
views
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 ...
- 748
7
votes
2
answers
586
views
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 ...
- 4,850
7
votes
2
answers
728
views
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 ...
- 2,126