**5**

votes

**1**answer

127 views

### Approximations of the identity on Lie groups and homogenous spaces

I'm looking for a nice (and preferably classic or book) reference for the following type of result:
Consider a transitive action of a compact Lie group $G$ on a compact manifold $M$ and a continuous ...

**1**

vote

**0**answers

89 views

### Subgroups of $GL(n,\mathbb{R})$ which are $Aut(L)$ for some Lie structure [closed]

What is a sufficient condition for a lie subgroup $G$ of $GL(n,\mathbb{R})$ to be the automorphism group of a Lie structure on $\mathbb{R}^{n}$. In particular does $O(n)$ satisfies this property?

**0**

votes

**0**answers

91 views

### Brauer characters of finite simple group $E_8(5)$

I would like to find the irreducible characters of the group $E_8(5)$ (mod 2)?
Can anyone help? (I am elementary in working with Brauer characters)
Many thanks

**8**

votes

**4**answers

310 views

### Automorphism group of flag manifolds?

If $F=F(n_1, \ldots, n_k)$ is the (complex) flag variety associated to the partition $(n_1, \ldots, n_k)$, what is the automorphism group of $F$? Here I mean holomorphic and/or variety automorphisms.
...

**6**

votes

**0**answers

126 views

### Zariski closure of orbits of real groups on complex flag manifolds

Let $G$ be a complex reductive algebraic group defined over $\mathbb R$, and $G_0$ its real points. Then the orbits of $G_0$ on $G/B$ need not be real algebraic subvarieties. Take $G=SL_2(\mathbb C)$, ...

**3**

votes

**1**answer

219 views

### Do cyclic product vectors generatating irreducible representation of a Lie group come from a unique orbit?

Consider a Hilbert space $\mathcal{H}$ which is a carrier space of a unitary, irreducible and strongly continuous representation $\Pi$ of a Lie group $G$. Let $\Pi\otimes \Pi$ denote the corresponding ...

**1**

vote

**0**answers

59 views

### Reference Help: Matsuki duality Orbits

I'm studying the Matsuki duality of $G_0$-orbits and $K$-orbits over a flag manifold $G/P$ where $G$ is semisimple complex Lie group and $P$ is a parabolic subgroup. I would like to study some ...

**3**

votes

**1**answer

203 views

### What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence?

If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, ...

**1**

vote

**1**answer

156 views

### The set of leaves of the distribution $D$ on coadjoint orbit $O_{\mu}$

Let $G$ be a compact connected Lie group and $O_{\mu}$ be a coadjoint orbit where $\mu\in \mathfrak{g}^*$ and $\mathfrak{g}^*$ is the dual of the Lie algebra of $\mathfrak{g}=\mathrm{Lie}(G)$. Let ...

**4**

votes

**1**answer

115 views

### connections on principal bundles over $S^1$

Suppose $G$ is a compact connected Lie group and $P$ is a $G$-bundle over $S^1$, $A$ is a connection. Then we can choose a frame such that $A = a d\theta$ where $a\in \mathfrak{g}$ is constant. My ...

**5**

votes

**1**answer

332 views

### Root space decomposition

What is the root system for the special unitary lie algebra $\mathfrak{su}(p, q)$. Remind that these are matrices of the form
$\left(
\begin{array}{cc}
X & Y \\
\overline{Y}^t & Z ...

**3**

votes

**1**answer

110 views

### Reference to definition of matrix log with domain SO(3) which is Borel measurable

I was trying to set up an inverse to matrix exponential $\exp:\mathrm{Skew}(3\times 3)\to SO(3)$ that "covers" the biggest possible domain and is Borel measurable. I was wondering if there is a ...

**4**

votes

**1**answer

419 views

### Topologie sur l'ensemble des sous-groupes de GL_n(R)

Bonjour,
Est ce qu'il existe une topologie naturelle sur l'ensemble des sous-groupes du groupe général linéaire ?
English translation: "is there a natural topology on the set of subgroups of the ...

**4**

votes

**0**answers

89 views

### Groups of operators between local unitaries and full unitaries

Consider the group $U(d_1) \otimes U(d_2)$ of "local unitary" operators acting on the complex space $\mathbb{C}^{d_1} \otimes \mathbb{C}^{d_2}$ (i.e., $U(d_1) \otimes U(d_2)$ is the group of unitary ...

**4**

votes

**1**answer

113 views

### Closed form for 3j-symbol ratios

I am working on the spherical harmonic decomposition of cosmic microwave background maps, therefore I often deal with functions that are proportional to Wigner 3J symbols/Clebsch–Gordan ...

**2**

votes

**0**answers

104 views

### On Eigenvalues of the symmetric linear transformation related to a lie algebra's representation?

Let $\mathfrak{g}$ be a quadratic (finite dimensional) lie algebra and $\rho:\mathfrak{g}\rightarrow \mathfrak{gl}(W)$ be an anti-symmetric representation of $\mathfrak{g}$ on a finite dimensional ...

**1**

vote

**0**answers

88 views

### Finding density of Haar measure relative to the Liouville volume measure on $\frac{G^{\mathbb{C}}}{P}$

Let $G$ be a connected compact Lie group and $G^{\mathbb{C}}$ be the complexification of the Lie group $G$ then we know that by polar decomposition we can write $G^{\mathbb{C}}\cong G\times ...

**2**

votes

**1**answer

139 views

### lifting group action

Let a Lie group $G$ act on a manifold $M$. Under which condition(s) we have a lifting action of $G$ on the universal covering of $M$?
By Bredon, I already know that there is a group $\widetilde{G}$, ...

**0**

votes

**1**answer

266 views

### Computing the fundamental group of a flag variety

Let $G$ be a compact and connected and simply connected Lie group and $\mathfrak{g}$ be its Lie algebra and $x\in\mathfrak{g}^*$. How can we compute the fundamental group of $G/G_x$ where $G_x$ is ...

**6**

votes

**5**answers

478 views

### Reference requested: Random walk on groups

I am looking for a good reference to learn about random walks on groups (either finite groups or Lie groups). Ideally, I would like a reference for general theory of random walks on groups that is ...

**1**

vote

**1**answer

227 views

### labeling state vectors in representation space of a simple lie algebra

Given a simple lie algebra (over ${\mathbb C}$ or ${\mathbb R}$). What is the number
of operators such that their eigenvalues sufficiently label all state vectors in the algebra's representation ...

**4**

votes

**1**answer

121 views

### Connection between degree of growth and return probabilities of random walks on Lie groups

Let $G$ be a finitely generated group of polynomial growth, let $\mu$ be a non-degenerate symmetric probability measure with finite support on $G$, and let $d$ be the degree of growth of $G$. ...

**2**

votes

**0**answers

353 views

### Mathematica package for Lie algebra computations?

I am interested in performing Lie algebra computations in Mathematica. I did a bit of searching and found several packages (LieART, KILLING, SuperLie, maybe more), and wondered if anyone would ...

**1**

vote

**0**answers

45 views

### Analytic vectors of elliptic elements of the enveloping algebra in a strongly continuous representation of a Lie group

Let $G$ be a semi-simple real Lie group, and $\rho$ a strongly continuous unitary representation of $G$ in a Hilbert space $H$. Let $\mathfrak{g}$ be its Lie algebra and $d\rho$ be the infinitesimal ...

**0**

votes

**1**answer

194 views

### Subgroups with trivial Centralizers

Let $G$ be a compact, simply-connected, and simple Lie group. Let $H$ be a closed subgroup of $G$ that has the same centralizer as the center of $G$. Is there a nice classification of such subgroups?
...

**4**

votes

**1**answer

282 views

### Tannaka–Krein duality

First I would like to stress that maybe I don't have a necessary background from the theory of Lie groups. I met the topic of Tannaka–Krein duality while reading the book of Gracia–Bondia, Varilly and ...

**4**

votes

**0**answers

85 views

### Adjoint orbit of two vectors

Let $G$ be a simple compact real Lie group and let $\mathfrak g$ be its Lie algebra. Let $u,v\in \mathfrak g$ be two distinct unit vectors and $H\subset \mathfrak g$ be a hyperplane with normal vector ...

**12**

votes

**0**answers

463 views

### How come Cartan did not notice the close relationship between symmetric spaces and isoparametric hypersurfaces?

Elie Cartan made fundamental contributions to the theory of Lie groups and their geometrical applications. Among those, we can list the introduction of the remarkable family of Riemannian symmetric ...

**1**

vote

**1**answer

194 views

### Finding the 2nd homotopy group $\pi_2(G^\mathbb{C}/P)$

Let $G$ be a compact connected and simply connected Lie group and $G^\mathbb{C}$ be the complexification of Lie group (with is diffeomorphic with $G^\mathbb{C}\cong T^*G$) then I am looking for ...

**5**

votes

**0**answers

391 views

### Connections on a Lie Group

A Lie group $G$ can be considered as a reductive homogeneous space in at least two different ways; $G/\{e\}$ and $G\times G/G^*$. In the first case, the canonical connection associated with the ...

**0**

votes

**0**answers

79 views

### lie groups and coset manifolds

Let $G$ be a lie group, $H\subseteq K\subseteq G$ be closed subgroups , and $H$ be normal in $G$. I wonder if the coset manifold $\frac{\frac{G}{H}}{\frac{K}{H}}$ is diffeomorphic to $\frac{G}{K}$.
...

**1**

vote

**2**answers

219 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 ...

**4**

votes

**5**answers

545 views

### How to characterize Dirac's gamma matrices in differential geometry?

I want to understand what is the interpretation of Dirac gamma matrices in differential geometry. Basically, I am considering the Dirac matrices as 3-indexed tensors, which means a tensor with 1 ...

**5**

votes

**2**answers

359 views

### When is a subgroup of a Lie group itself a Lie group?

Assume that $G$ is a Lie group. Is it understood which subgroups of $G$ are Lie groups?
Ideally, I would like to make no extra assumption about $G$. In particular, $G$ can be infinite dimensional.
...

**2**

votes

**1**answer

102 views

### Subgroups of $E(n) = \mathbb{R}^n \rtimes O(n)$ with trivial orbit space

Let G be a subgroup of $E(n) = \mathbb{R}^n \rtimes O(n)$(the rigid motions of $\mathbb{R}^n$ ) with orbit space as a point.
Example: the group of all translations of $\mathbb{R}^n$ and of course any ...

**1**

vote

**1**answer

232 views

### $SO(N^2-1)$ and the adjoint representation of $SU(N)$

It is a known fact that the adjoint representation of $SU(N)$ is a proper subgroup of $SO(N^2-1)$.
I would like to know how a generic $(N^2-1)\times (N^2-1)$ special ($det =1$), orthogonal matrix $O$ ...

**2**

votes

**1**answer

167 views

### Indefinite orthogonal groups over p-adics

Let $q$ be a rational quadratic form. How can we think of a Cartan decomposition of $O_q(Q_p)$? Is there a notion of Cartan involution for p-adic field, so that we can execute same process as we do ...

**1**

vote

**1**answer

147 views

### Each coadjoint orbit of a compact connected Lie group $G$ admits a $G$-invariant generalized complex structure

I am looking for a proof or counterexample for following assertion
Each coadjoint orbit of a compact connected Lie group $G$ admits
a $G$-invariant generalized complex structure (In sense of ...

**6**

votes

**1**answer

248 views

### Can Galois conjugates of lattices in SL(2,R) be discrete?

Let $\Gamma$ be a lattice in $SL(2,\mathbb{R})$. Suppose that the trace field of $\Gamma$ is a totally real number field of degree $d$. This gives $d$ homomorphisms $\rho_i:\Gamma\to SL(2,\mathbb{R})$ ...

**3**

votes

**1**answer

162 views

### Stationary curves on homogeneous spaces

Consider $M \cong G/K$ ($G$ a lie group with a transitive action on $M$ and $K$ a subgroup) and consider a Lagrangian $\mathcal{L}: TM \rightarrow \ \mathbb{R}$ (no time dependence). Consider also ...

**6**

votes

**0**answers

269 views

### Injectivity of Lie group exponential function

If $G$ is a (finite-dimensional) Lie group, then the exponential function $\exp\colon\mathfrak{g}\to G$ is injective on some identity neighbourhood. If, moreover, $\mathfrak{g}$ is semi-simple and ...

**5**

votes

**2**answers

267 views

### Summary of Lie-Algebra integration tactics

If this is in the scope of MO, I would like to gather here the known tactics of
Lie algebra integration, since it appear surprisingly hard to find such a
compendium, library or any other kind of ...

**0**

votes

**2**answers

171 views

### A question on Lie algebras

To what extent, the following types of Lie algebras are classified :
Those Lie algebras $L$ such that every Lie Group $G$ with $Li(G)\sim L$, is necessarily compact.

**2**

votes

**1**answer

160 views

### Distance measure for noisy $SE(3)$ transforms

I have a transformation $T \in SE(3)$ parameterized by a mean quaternion $q$ with covariance matrix $\Sigma_q \in R^{4\times4}$ and a mean translation $t \in R^3$ with covariance matrix $\Sigma_t \in ...

**6**

votes

**1**answer

207 views

### Flag manifolds (=R-spaces): quotients by parabolic subgroups vs. isotropy representation

Real flag manifolds (also known as R-spaces) can be defined in two ways which I believe are equivalent although some fine print may have escaped me:
as a quotient of a semisimple real Lie group $G$ ...

**4**

votes

**0**answers

124 views

### Classification of Compact Symplectic Homogeneous Spaces

Let $M=G/H$ be a compact homogeneous space, $G$ a compact Lie Group and $H$ a closed subgroup. Is there some classification, akin to the Kaehler case, for which such manifolds admit a symplectic ...

**6**

votes

**1**answer

380 views

### Origin of symbols used for half-sum of positive roots in Lie theory?

The Weyl character formula is a central result in the finite dimensional representation theory of semisimple Lie groups, algebraic groups, Lie algebras. Related questions on MO include these here ...

**0**

votes

**0**answers

118 views

### A question on lie groups( Lie algebras)

What is an example of a compact Lie group $G$ which is not isomorphic to a product of two nontrivial lie groups and satisfies the following property:
There are two non zero vector fields $X, Y \in ...

**1**

vote

**1**answer

68 views

### Hermitian symmetric structure on a homogeneous subspace

Let $G$ be a semisimple group over $Q$ and $K$ a maximal compact subgroup of $G(R)^+$.
I am assuming that $G(R)^+/K$ has a structure of a non-compact Hermitian symmetric domain.
Let $g= p + k$ be ...

**3**

votes

**0**answers

95 views

### Equivariant Poincare Series of Based Loop Group of SU(2)

Let $\Omega SU(2)$ denote the based loop group of $SU(2)$, and consider the action of $S^1$ on $\Omega SU(2)$ as a maximal torus of $SU(2)$. (This is not the "loop rotation" action.) Is there an ...