# Tagged Questions

**9**

votes

**1**answer

240 views

### Decomposition of $\mathrm{O}(n)$-modules coming from differential geometry

Let $V$ be a $n$-dimensional real vector space equipped with a positively definite scalar product $g$ and let $\mathrm{O}(n)$ be the automorphism group of $(V,g)$. View $V^{\otimes k}$ as a ...

**5**

votes

**1**answer

290 views

### Is there a specific geometric meaning why fractional charges are allowed in SU(N) gauge theories?

So in the standard model of particle physics, there exist particles with fractional charge. What this means geometrically is as follows: We are given a smooth manifold with a principal $U(1)$ bundle ...

**1**

vote

**0**answers

161 views

### Cotangent bundle of symmetric space is symmetric space?

Let $G$ be a connected Lie group. Then a symmetric space for $G$ is a homogeneous space $G/H$ where the stabilizer $H$ of a typical point is an open subgroup of the fixed point set of an involution ...

**8**

votes

**1**answer

240 views

### Extended Deformation Theory (dg-Lie algebra principle in positive characteristic?)

Recently, I looked at articles that make use of Deligne's idea that "in characteristic 0 every deformation problem is governed by a differential graded
Lie algebra" as explained first in ...

**1**

vote

**2**answers

212 views

### When representation of two different coadjoint orbits are equivalent?

Let $G$ be a compact connected Lie group and $\mu:T\to S^1$ be a representation of a maximal torus $T \subset G$ and $\lambda=d\mu$ be a weight for some $\lambda\in\mathfrak{t}^*$ (where ...

**1**

vote

**0**answers

186 views

### quantization for integral coadjoint orbits

I am looking for a referrence for proof of following statement.
Let $G$ be a semi-simple Lie group and $\mathfrak{g}$ be its Lie algebra and $\alpha_k$ be simple roots of $\mathfrak{g}$ and $\mu_0$ ...

**6**

votes

**2**answers

244 views

### Whitney stratification and affine grassmanian

Let $G$ a simply connected group over $\mathbb{C}$ and $Gr:=G(\mathbb{C}((t)))/G(\mathbb{C}[[t]])$ the affine grassmannian. By Cartan decomposition we have a partition of stratas indexed by ...

**1**

vote

**1**answer

424 views

### Para-Complexification of Lie Groups

Let $G$ be a real Lie group. Then the complexification $G_\mathbb{C}$ of $G$ is the unique complex Lie group equipped with a map $φ:G\to G_\mathbb{C}$ such that any map $G\to H$ where $H$ is a
...

**1**

vote

**0**answers

127 views

### How to pick out harmonics based on boundary conditions?

(..this is almost a continuation of my last question (which got closed!)...) Let me first rewrite one of the main results of this paper, http://calvino.polito.it/~camporesi/JMP94.pdf in a coordinate ...

**5**

votes

**1**answer

220 views

### Spin manifolds with one parallel spinor

Are there any examples of D-dimensional Ricci-flat Riemannian (spin) manifolds of dimension D= 2,3,4,5 with the dimension of the space of parallel spinors equal to 1? And the same question for the ...

**2**

votes

**2**answers

229 views

### The Szego projector, the dual disc bundle $\overline{D}$ and representation of $S^1$ on $H^2$($\overline{D}$)

This construction arises when constructing the Szego projector.
Let's consider the dual disc bundle $\overline{D}$ of a positive Hermitian line bundle ($L$,$h$) over a compact Kahler manifold $M$, ...

**2**

votes

**1**answer

534 views

### Kirillov-Kostant-Souriau Theorem on $\mathfrak{g}\oplus \mathfrak{g^*} $

My question is about the extention of kirillov's symplectic structure on coadjoint orbits. The most remarkable feature
of the coadjoint representation is the fact that all coadjoint orbits possess a
...

**3**

votes

**0**answers

150 views

### differential geometry of Limaçons

While reading up on Limaçons I learned they are the envelope of one circle around another.
The limaçon can be generated by specifying a fixed point P, then
drawing a sequences of circles with ...

**2**

votes

**1**answer

117 views

### The measure on the harmonic spectrum from Selberg trace formula

One can see the following two equations,
Theorem 6.1 (Selberg Trace formula) on page 26 of these notes.
Equation 3.19 and 3.20 on page 11 of this paper.
I vaguely feel that these two are the ...

**10**

votes

**2**answers

300 views

### Invariant differential operators on real Grassmannians

I am looking for an explicit description of the algebra of $SO(n)$- or, better, $O(n)$-invariant differential operators on the real Grassmann manifolds of $k$-dimensional linear subspaces in the ...

**0**

votes

**0**answers

230 views

### Local System and Gauss-Manin connection

Fix a complex manifold $X$. Then if we have a line bundle $L=\mathcal{O}(D)$ together with Gauss-Manin connection $\nabla: L \rightarrow L \otimes \Omega^{1}_X$, we get the locally constant sheaf $F$ ...

**5**

votes

**4**answers

334 views

### Prequantization and Hilbert space

In the definition of pre-quantization on a symplectic manifold $(M,\omega)$, we represent a function $f\in C^{\infty}(M)$(with Lie algebra structure) to $\hat{f}$ in the Hilbert space $L^2(M,L,\mu)$ ...

**1**

vote

**0**answers

148 views

### Equivariant $K$-theory, singular vectors, and flag manifolds

For a homogeneous space $M = G/B$, with $G$ a (complex) semi-simple Lie group, it is very well-known that equivariant vector bundles $E$ over $M$ correspond to representations $(V_{\lambda},\lambda)$ ...

**2**

votes

**1**answer

253 views

### Laplacian on coset spaces

Edited after @J. Martel's comment: Let us consider the sphere $S^n$ (embedded in $\mathbb{R}^{n+1}$). We know that if $X_i$ represent the vector fields on $S^2$ giving the rotation about the ...

**0**

votes

**1**answer

127 views

### A question about G-Manifolds

I am looking for a clear reason for following fact:Is there any reference ?
Why a $G$-invariant differential form $\omega$ on a homogeneous $G$-manifold $M=G/H$ is uniquely determined by its value at ...

**1**

vote

**1**answer

78 views

### $SU(n)$-invariant subring of $\Lambda^{*}\mathbb{R}^{2n}$

I have the following question: Let $R \subset \Lambda^{*}\mathbb{R}^{2n}$ be the sub-ring of forms which are preserved by $SU(n)$. How can one show that this subring is generated by $\Omega_{0}$ and ...

**3**

votes

**1**answer

384 views

### Differential equations and Lie groups

I am a physicist and I am pondering over a particular generalization of Stokes' theorem and Maxwell's equations. They apply to vector fields like the electric or magnetic one. However if the vectors ...

**1**

vote

**0**answers

133 views

### Relationship between stabilizers of a general point and a boundary point

Let $V$ be an n-dimensional complex vector space, and $u\in S^nV$ be a polynomial, $G(u)$ be the stabilizer of $u$ in $GL(V)$. Let $[v]\in\overline{GL(V)\cdot[u]}\subset\mathbb{P}(S^nV)$, but $v\notin ...

**22**

votes

**3**answers

808 views

### Rep Theory Consequences of Bott--Weil--Borel

I've been getting interested in the (Bott--)Borel--Weil theorem lately. As a (mainly) geometer it is very interesting to see representation appearing (from nowhere as far as I can see) in the theory ...

**5**

votes

**0**answers

196 views

### Group Representations and Holomorphic Vectors Bundles over Homogeneous Spaces (extending Borel--Weil)

For a flag manifold $F$ of a group $G$, the Borel--Weil theorem deals with representations of $G$ on the holomorphic sections of the line bundles over $F$.
Let us consider a general framework than ...

**5**

votes

**1**answer

261 views

### Representations of SO(3) and vector bundles on BSO(3)

Let $V$ be the vector bundle over $BSO(3)$ associated to the adjoint representation of $SO(3).$ Then $V$ does not have a nonzero section. One way to see this is that the Steifel-Whitney class ...

**1**

vote

**2**answers

287 views

### Automorphy Factors and Bundles

The question I'm considering is the following: given an 1-cocycle in of the modular group in Hom$(H;\textrm{GL}_{r}(C))$ call it $f$ when does it induce a vector bundle structure on the corresponding ...

**3**

votes

**2**answers

527 views

### Irreducible representation decomposition of tensor on manifold with metric

I'm aware that for some tensor product space, Schur-Weyl duality lets me decompose the space into irreducible representations by looking at irreps of the symmetric group. The simplest example is ...

**11**

votes

**1**answer

451 views

### A question on invariant theory of $GL_n(\mathbb{C})$.

Let $\rho$ denote the irreducible algebraic representation of $GL_n(\mathbb{C})$ with the highest weight $(2,2,\underset{n-2}{\underbrace{0,\dots,0}})$.
Let $k\leq n/2$ be a non-negative integer. ...

**11**

votes

**1**answer

267 views

### Most degenerate Weyl tensors in Riemannian and Lorentzian signature

Hi all,
This is my first post on Math Overflow! I've been stuck on the following question and was wondering if anyone might have any insight on it. Here it is:
Let $n \geq 5$. Let $G = SO(n)$ or ...

**3**

votes

**1**answer

732 views

### Representation theory of (anti)self-dual tensors

I am using usual physics notations and I guess the physics motivations of this question are obvious.
Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the Lie algebra be, ...

**30**

votes

**5**answers

2k views

### Beautiful descriptions of exceptional groups

I'm curious about the beautiful descriptions of exceptional simple complex Lie groups and algebras (and maybe their compact forms). By beautiful I mean: simple (not complicated - it means that we need ...

**13**

votes

**2**answers

795 views

### Regarding Cayley Graphs of Property (T) Groups

A well-known application of Kazhdan's Property (T) is the construction of expander graphs. Background on this is discussed, for example, in this post on Terry Tao's blog. Essentially, Cayley graphs of ...

**4**

votes

**2**answers

293 views

### Finite groups admitting free isometric actions on round spheres

Is there some sort of classification of finite groups $G$ such that for at least one $n$ the group $G$ admit a free isometric action on the standard sphere $S^n $of curvature 1? Are there some simple ...

**4**

votes

**1**answer

376 views

### Associated vector bundles of infinite rank and induced connections

Let $\mathbb{V}$ be a representation of a Lie group $G$ and let $P \to M$ be a principal $G$-bundle with a principal connection. If $\mathbb{V}$ is finite-dimensional, then one can associate to this ...

**12**

votes

**1**answer

378 views

### General Isoperimetric Inequality via Representation Theory of SO(n)

Is there a known proof of the $n$-dimensional isoperimetric inequality which generalizes Hurwitz's proof using Fourier analysis in the $2$-dimensional case?
Specifically, I imagine such a proof would ...

**3**

votes

**1**answer

655 views

### first chern class and spin structures

Let M be a compact complex manifold. Then is it true that if the first Chern class of M is even, then M admits a spin structure?

**5**

votes

**1**answer

313 views

### Splitting principle in equivariant cohomology

The following is a weaker version of what is called splitting principle in
Appendix C, page 12, see also for a lighter version Brions Eq cohom and eq intersection theory, page 6:
Let $G$ be a compact ...

**5**

votes

**2**answers

1k views

### Why SU(3) is not equal to SO(5)?

I am asking in the sense of isometry groups of a manifold. SU(3) is the group of isometries of CP2, and SO(5) is the group of isometries of the 4-sphere. Now, it happens that both manifolds are ...

**7**

votes

**1**answer

504 views

### Spinor space to Euc. vector space: does there exist a universal bilinear map?

Let $S$ be a spin representation of the Euclidean
spin group $Spin(d)$ and let ${\mathbb R}^d$
be Euclidean $d$-space with $Spin(d)$ action on it
in the canonical way, via the 2:1 cover to $SO(d)$.
...

**15**

votes

**7**answers

3k views

### Topology of SU(3)

$U(1)$ is diffeomorphic to $S^1$ and $SU(2)$ is to $S^3$, but apparently it is not true that $SU(3)$ is diffeomorphic to $S^8$ (more bellow). Since $SU(3)$ appears in the standard model I would like ...

**3**

votes

**2**answers

528 views

### Spinors on orbifolds

Let $R^{n}$ be a cone over sphere $S^{n-1}$ with the metric $g = dr^2 + r^{2}g[S^{n-1}]$ ($r> 0$).
Whether it is true that the cone over $S^{n-1}/Z_{2} = RP^{n-1}$ has twice less parallel spinors, ...

**7**

votes

**2**answers

382 views

### Symmetric spaces, Horocycle spaces and intertwining operators

Let $G=KAN$ be an Iwasawa decomposition of a connected semisimple Lie group with finite center. Let us assume for simplicity that the associated symmetric space $G/K$ has rank 1.
Harish-Chandras ...

**4**

votes

**1**answer

735 views

### Classification of discrete subgroups of the unitary group

Let $U(n)$ be the unitary group. From André Weil's paper "On discrete subgroups of Lie groups" it is well known that discrete cocompact subgroups of $U(n)$ have only a finite number of generators and ...

**2**

votes

**0**answers

337 views

### Diagram Calculus for Spinors

In quantum field theory class, we are doing spinors. The representations of the Lorentz algebra $\mathfrak{so}(3,1) = \mathfrak{su}(2) + i \mathfrak{su}(2)$ are indexed by two integers. Then (1,2) ...

**6**

votes

**4**answers

1k views

### Quotient space of $\mathbb{C}^5$ under the action of $SL(2,\mathbb{C})$

One sees that given the $SL(2,\mathbb{C})$ action on $\mathbb{C}^5$, thought of as the space of polynomials of the form,
$$a_0 x^4 + 4a_1 x^3 y + 6a_2x^2y^2 + 4a_3xy^3 + a_4 y^4$$
the ring of ...

**0**

votes

**1**answer

568 views

### Action of $SL(2,\mathbb{C})$ on representations of $SU(2)$

I want to precisely understand in what sense is (if it is!) $SL(2,\mathbb{C})$ the "complexified" version of $SU(2)$?
Can I think of it like choosing a natural matrix basis of the real three ...

**20**

votes

**3**answers

1k views

### When are the eigenspaces of the Laplacian on a compact homogeneous space irreducible representations?

I was writing up some notes on harmonic analysis and I thought of a question that
I felt I should know the answer to but didn't, and I hope someone here can help me.
Suppose I have a compact ...

**14**

votes

**4**answers

1k views

### Poincare dual in equivariant (co)homology?

Let $G$ be a compact Lie group, $X$ be a (compact, oriented) smooth manifold, with $G$ acts on $X$ smoothly. Then we can talk about the $G$-equivariant homology and cohomology.
My question: In what ...

**5**

votes

**4**answers

1k views

### Which Riemannian manifolds admit a finite dimensional transitive Lie group action?

This is a basically an adjusted version of my earlier question about how to define a convolution algebra on a general Riemannian manifold. The motivation for asking such a question of course comes ...