All Questions
Tagged with differential-operators lie-groups
13 questions
8
votes
0
answers
112
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Euler number of the complex of basic forms
Let $G$ be a compact Lie group and $\pi:P \to M$ a principal $G$-bundle. I would like to understand the geometry of $M$ through $P$ with the $G$-action.
I am trying to understand the Hopf bundle ($G=...
- 181
7
votes
1
answer
601
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Universal enveloping algebra and the algebra of invariant differential operators
Let $G$ be a Lie group and $\mathfrak{g}$ be its Lie algebra. Then $\mathfrak{g}$ may be interpreted as the Lie algebra of right (equivalently left) invariant vector fields. Let $\mathcal{U}(\mathfrak{...
- 9,330
7
votes
1
answer
535
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Diffeomorphisms on a real manifold whose derivative are holomorphic maps on the tangent bundle
Edit: According to the answers to the linked MSE question and the comment of Holonomia, I understand that the answer to the second question is that " Every tangent bundles is a complex ...
- 356
7
votes
0
answers
217
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Correspondence between Verma module morphisms and invariant differential operators - is it exact?
For a (complex, connected, simply connected, simple) Lie group $G$, a parabolic subgroup $P \subseteq G$, and a $\mathfrak g$-integral $\mathfrak p$-dominant weight $\lambda$, we can construct ...
- 1,368
5
votes
0
answers
203
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Differential equation involving Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
Throughout this post, we let $G$ denote the Lie group $\mathrm{SL}(2,\mathbb{R})$. For $t,\theta \in \mathbb{R}$ we define the following elements of $G$:
\begin{align*} A(t) &= \begin{bmatrix}e^t &...
- 1,769
4
votes
0
answers
114
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Mean-value type property for eigenfunctions of Casimir operator on $\operatorname{SL}(2,\mathbb{R})$
The purpose of this post is to ask for the community's opinion on a conjecture about a certain mean-value property for functions on $\operatorname{SL}(2,\mathbb{R})$. This conjecture appears at the ...
- 1,769
4
votes
0
answers
197
views
$G$-Invariant Differential Operators
Let $G$ be a complex algebraic group, $K$ a closed subgroup so that $X=G/K$ is a homogeneous space.
Let $\mathcal{D}(X)$ denote the algebra of differential operators on $X$. The group $G$ acts on $\...
- 1,077
4
votes
0
answers
134
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Spin equivariance of the Dirac operator-flat case
This question was posed on Math.SE but no one has answered it; it may be suitable for MathOverflow.
Let $D$ be the Dirac operator on $\mathbb{R}^n$ i.e. $D=\sum_{j=1}^nE_j\frac{\partial}{\partial ...
- 9,330
4
votes
0
answers
173
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Ring of SO(n)-invariant differential operators on M_n,m
I'm reading through Stephen Gelbart's paper "A Theory of Stiefel Harmonics." (http://www.ams.org/journals/tran/1974-192-00/S0002-9947-1974-0425519-8/).
There comes a point in the paper (Lemma 2.8) ...
- 41
3
votes
0
answers
129
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Differential operators on a compact Lie group associated to bracket-generating sets
Let $G$ be a compact connected Lie group of dimension $m$ with Lie algebra $\mathfrak g$.
Let $\{X_1,\dots,X_h\}$ be a linearly independent set of $\mathfrak g$.
Assume that $\{X_1,\dots,X_h\}$ is ...
- 2,446
2
votes
1
answer
90
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Germs of left invariant differential operators on a group
Are there germs at the identity of linear differential operators on a group which are not germs at the identity of left invariant differential operators?
I feel like the answer is no but the statement ...
user525442
2
votes
0
answers
55
views
Function annihilated by ideal in universal envelope
Let $G$ be a Lie group, $U(G)$ its universal enveloping algebra over $\mathbb C$ and let $J\ne U(G)$ be a left ideal. We consider $U(G)$ as the algebra of left-invariant differential operators on $G$.
...
user130903
2
votes
0
answers
429
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A Generalized De Rham cohomology
Edit According to the comment of Alex Degtyarev, I deleted the last part of the previous version.
Let $E$ be a real vector space. The complex valued $k$- tensors on $E$ is denoted by $L_{\mathbb{C}}^{...
- 356