**26**

votes

**1**answer

526 views

### Why do the adjoint representations of three exceptional groups have the same first eight moments?

For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...

**5**

votes

**1**answer

256 views

### If $G$ is absolutely simple simply connected, why is G(F_v) quasisimple for almost every valuation v?

Let $G$ be an absolutely simple simply connected and connected algebraic group defined over a global field $k$ with ring of integers $\mathcal{O}$. Fix an embedding of $G$ into $GL_n$. Given $v$ a non-...

**11**

votes

**3**answers

396 views

### $A_{\infty}$-structure on closed manifold

Is there an exmaple of a closed smooth connected manifold $M$ having a structure of $A_{\infty}$-space (with unit) but $M$ is not homeomorphic to a compact connectd Lie group as space ?
Edit: First, ...

**1**

vote

**0**answers

91 views

### Roots of matrices in $G_2(Z)$

Let $G_2$ denote the exceptional Lie group $G_2$ as a $\mathbb{Q}$-algebraic group. Suppose that is also given a matrix representation $\rho : G_2\rightarrow SO(7)$. Let $M$ be a matrix with integral ...

**6**

votes

**3**answers

303 views

### Simple lie algebras, (almost-)simple groups of Lie type

Take an algebraic group $G$ defined over a finite field $K$. Suppose its Lie algebra $\mathfrak{g}$ is simple. It should follow that $G$ is almost-simple. (By this I mean not that $G(K)$ is simple -- ...

**4**

votes

**0**answers

171 views

### Compactly supported distributions as a projective G-module

For a Lie group $G$ and a locally convex space $V$ let $\mathcal{E}(G,V)$ be the locally convex space of smooth functions from $G$ to $V$, and accordingly $\mathcal{E}_c^\prime(G,V)$ the space of ...

**0**

votes

**1**answer

66 views

### Points with finite stabilizer in Hamiltonian torus actions

Atiyah-Guillemin-Sternberg theorem asserts that the image of the moment map $\mu$ for a Hamiltonian $(S^1)^m$-action on a smooth compact symplectic manifold $(M^{2n},\omega)$ is a convex polytope of $\...

**3**

votes

**1**answer

80 views

### Uniform lattice in semidirect product

A uniform lattice in a locally compact group $G$ is a discrete subgroup $\Gamma\subset G$ such that $G/\Gamma$ is compact.
My question is whether a uniform lattice exists in the group
$$
G={\mathbb R}^...

**2**

votes

**0**answers

136 views

### Lie Algebra of Aut(GL(n,R))

What is the Lie Algebra of $Aut(Gl(n,F))$ when $F$ is either $\mathbb{R}$ or $\mathbb{C}$?
Is it enough to consider the injection via Hochschild:
$Aut(GL(n)) \to Aut(\mathfrak{gl}(n))$?
Edit: The ...

**2**

votes

**0**answers

149 views

### Exponential analogue of formal connections

Let $F=\mathbb{C}((t))$. Let $G=GL_n$. Then $G(F)$ acts on $\mathfrak{g}(F)$ by gauge transformation:
$$
g.x:=gxg^{-1} + \dot{g}g^{-1},\quad \quad g\in G(F), \quad x\in \mathfrak{g}(F).
$$
Here, $\...

**1**

vote

**2**answers

138 views

### Four Sphere Fibrations

Does there exist a manifold $M$ and a compact Lie group $H$ such that we have a fibration $H \to S^4 \to M$, where $S^4$ is the four sphere?

**2**

votes

**0**answers

143 views

### Deligne-Simpson problem for classical groups

Additive Deligne-Simpson problem was partially prooved by Kostov. Also there is Crawley-Boevey's approach to the question. The problem is about existence of a solution of the equation
$$
A_1 +...+A_n =...

**2**

votes

**0**answers

88 views

### Set of Special Unitary Matrices that are dense in SU(4) and obey certain relations

I'm trying to find a finite set of 4x4 Unitary matricies $\{U_1,U_2,\ldots U_N\}$ such that the matrices are dense in SU(4), and obey the relations:
$[U_i, U_j] = 0$ for $|i-j|>1$
$U_iU_{i+1}U_i=...

**2**

votes

**0**answers

98 views

### Lie Symmetries of the Bessel Differential Equation

The Bessel differential equation has an arbitrary looking form, but a lot is known about it.
$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$
Is there a way to derive the Bessel ...

**4**

votes

**2**answers

278 views

### odd length Chevalley relations (in rank two)

The unipotent radicals $\text{N}$ of the Borel subgroups of the complex algebraic groups of type $A_2$, $B_2$, and $G_2$ can each be abstractly presented using two one-parameter subgroups $x_1, x_2: \...

**0**

votes

**0**answers

47 views

### Generalized Gaussian Decomposition

Let $G$ be a connected complex semisimple Lie group. Let $H$ be a maximal torus of $G$, let $W$ be the Weyl group of $G$, and let $N_\pm$ be a pair of opposite maximal unipotent subgroups. For each ...

**4**

votes

**0**answers

158 views

### Representation theory and associated bundles

I am looking for a text or set of notes that discusses the relationship between the representation theory of Lie groups and associated vector bundles, preferably using modern categorical language. For ...

**8**

votes

**1**answer

334 views

### Intuition for the Cartan connection and “rolling without slipping” in Cartan geometry

Consider a Cartan geometry $\pi: \mathcal{G} \to M$ with Cartan connection $\omega$ modelled on the Klein geometry $(G, H)$.
The Cartan connection is supposed to formalize what it means to "roll ...

**1**

vote

**0**answers

61 views

### Kirillov orbit Method for Complex nilpotent groups

Let $G$ be a nilpotent simply connected real Lie group. From the classical work of Kirillov, it is well-known that the irreducible unitary representations of $G$ are in a canonical bijective ...

**6**

votes

**0**answers

94 views

### Samelson Products in $SO(n)$

Given a topological group $G$ one forms the commutator $c\colon G\times G\rightarrow G$, $(x,y)\mapsto xyx^{-1}y^{-1}$. This map then factors through the smash $G\wedge G$. This map is the most ...

**17**

votes

**2**answers

654 views

### Solving equations in SO(3) : an open problem by Jan Mycielski

I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group? (The American Mathematical Monthly, 1977, http://www.jstor.org/stable/...

**20**

votes

**4**answers

918 views

### Technical issue in the approach to Lie groups taken in a book

I'm teaching Lie groups and Lie Algebras out of Brian C. Hall's book (Lie Groups, Lie Algebras, and Representations: An Elementary Introduction, Springer), which I've enjoyed using. I'm confused about ...

**3**

votes

**1**answer

197 views

### Totally geodesic subgroups in Lie groups

Let $G$ be a Lie group with a left invariant metric $g$.
Let $H$ be a (closed) Lie subgroup of $G$, and assume $g$ is right-$H$-invariant. (That is $d(R_h)_e:T_eG \to T_hG$ is an isometry for every $...

**9**

votes

**3**answers

276 views

### Construct discrete series of SL(2,R) as kernel of twisted Dirac operators

I’m studying the paper of (Baum-Connes-Higson, ex 4.25), and I would like to give an explicit computation for the Connes-Kasparov conjecture for SL(2,R).
The idea is that each non-trivial ...

**6**

votes

**0**answers

140 views

### An analogue of Deligne--Lusztig theory for real groups?

I am considering the following analogue of Deligne--Lusztig theory:
Take e.g. $G=GL_n(\mathbb{C})$, and let $F$ be the complex conjugate, then we have
$G^F=GL_n(\mathbb{R})$. Consider the ``Lang map''...

**4**

votes

**1**answer

145 views

### Lie functor preserves “surjections” in synthetic differential geometry?

In classical finite-dimensional differential geometry, the Lie functor preserves surjections, sending a surjective Lie group homomorphism to a surjective Lie algebra homomorphism.
As pointed out ...

**3**

votes

**1**answer

98 views

### Weingarten function for unitary group

Studying integration over unitary group I came across this function, the Weingarten function Wg, such that
$$ \int_{\mathcal{U}(N)} \prod_{k=1}^{n} U_{i_kj_k}
U^*_{m_k r_k} dU=\sum_{\tau,\sigma\in S_n}...

**8**

votes

**2**answers

227 views

### Quasi-isometric rigidity of Nil

Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...

**8**

votes

**2**answers

528 views

### Interpret Fourier transform as limit of Fourier series

Let $V=\mathbb{R}^n$, $\Lambda_r=2\pi r \mathbb{Z}^n \subset V (r>0)$ a lattice; $V^*\cong\mathbb{R}^n$ the dual vector space of $V$, and $\Lambda_r^*=\frac{1}{2\pi r} \mathbb{Z}^n =\text{Hom}(\...

**0**

votes

**1**answer

92 views

### Maximal split torus of universal chevalley group

Let $G$ be simply connected chevalley group over a field $K$. I am following the notations as in 'Lectures on Chevalley group' by Steinberg (Yale lectures). Let $H$ be the subgroup generated by $\{h_{\...

**10**

votes

**2**answers

238 views

### Can one describe the multiplication of two Bruhat cells?

For $G$ a simple linear algebraic group and $B$ a fixed Borel subgroup, we have the Bruhat decomposition $G = \coprod_{w \in W} B\dot{w}B$, where $W$ is the Weyl group and $\dot{w}$ is any ...

**5**

votes

**1**answer

163 views

### The action of $GL_{\infty}$ on the infinite wedge space

This is a question from the book "Highest weight representations of infinite dimensional Lie algebras, 2nd ed" by V. G. Kac, A. K. Raina, and N. Rozhkovskaya.
Consider the following objects:
the ...

**16**

votes

**5**answers

556 views

### Is there a formula for the Frobenius-Schur indicator of a rep of a Lie group?

Let $G$ be a simple algebraic group group over $\mathbb C$.
Let $V$ be a self-dual representation of $G$.
Let $\lambda$ be the highest weight of $V$.
Write $\lambda$ as a sum of fundamental weights: $...

**4**

votes

**0**answers

123 views

### When can a locally compact group be approximated by discrete subgroups?

This question is about partitioning a (locally) compact group into cells by using discrete subgroups.
Let $G$ be a locally compact group. (I am really most interested in the case where $G$ is a ...

**2**

votes

**0**answers

100 views

### Are singular critical points isolated for control systems on compact semisimple Lie groups

Given a control system on $\mathrm{SU}(n)$ (or any other compact, semi-simple Lie group I suspect) of the form:
$\frac{d U_t}{dt} = (A + w(t)B)U_t$
where $A,B \in \mathfrak{su}(n)$ generate the ...

**3**

votes

**1**answer

91 views

### Characterization of $L^1(\text{SL}(3,\mathbb{R}))$ [closed]

Is there a characterisation of the integrable functions on SL($3,\mathbb{R}$) or an explicit expression for the Haar measure?

**2**

votes

**0**answers

167 views

### Manifolds as simultaneous coset spaces

Let $X$ and $Y$ be real manifolds. Under what conditions is there a Lie group $G$ with transitive actions on $X$ and $Y$ such that the stabiliser of a point in $X$ is a subgroup of the stabiliser of ...

**0**

votes

**0**answers

66 views

### G-invariant functions on manifold for G compact

In a paper I saw the following statement:
Let $M$ be a connected symplectic manifold and $G$ be a compact Liegroup acting symplectically and hamiltonian on $M$. Let $\Phi \colon M \to \mathfrak{g^*}$ ...

**3**

votes

**2**answers

155 views

### Commutator 2-forms on Lie groups

Let $G$ be a compact Lie group and $\mathfrak g$ its Lie algebra.
For any $f$ in the dual space $\mathfrak g^*$, we can define a skew-symmetric bi-linear form on $\mathfrak g$ by $(A,B)\mapsto f([A,B])...

**1**

vote

**1**answer

110 views

### Concept of Facets in the structure of reductive algebraic groups

Where can I find a precise definition of Facet ? In some online notes it is stated that Facet is a maximal subset of co-characters having the same sign for every root. But shouldn't then every facet ...

**4**

votes

**1**answer

80 views

### $E \times_H \mathbb{R}^n$ is isomorphic to the total space of the tautological bundle $\gamma^n$ over $G_n(\mathbb{R}^{n+k})$?

Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...

**7**

votes

**1**answer

105 views

### $\text{GL}(n + k, \mathbb{R})$ is a principal $H$ bundle over the Grassmann manifold $G_n(\mathbb{R}^{n+k})$?

Let $H$ be the subgroup of $\text{GL}(n + k, \mathbb{R})$ consisting of matrices whose lower $n \times k$ block is empty; i.e. consisting of matrices of the form$$\begin{pmatrix} A & * \\ 0 & ...

**2**

votes

**2**answers

69 views

### Abelian isometry groups of codimension one

Good day.
Let (M,g) be an n-dimensional Riemannian manifold (complete, if you wish), and suppose that there exists an n-1 dimensional Abelian group acting by isometries on M. Or locally, near a point ...

**0**

votes

**0**answers

69 views

### Rigidity of lower-dimensional lattices in Euclidean groups

Informal intro / motivation:
Suppose I have an infinite set of atoms arranged in a 2D periodic crystalline "sheet". By crystalline I simply mean that it is preserved by the action of integer linear ...

**15**

votes

**2**answers

472 views

### Exotic smooth structures on Lie groups?

If a topological group $G$ is also a topological manifold, it is well-known (Hilbert's 5th Probelm) that there is a unique analytic structure making it a Lie group.
However, for a compact Lie group $...

**0**

votes

**1**answer

124 views

### A particular Lie algebra $L_{n}$ and (various) lie groups whose Lie algebra is isomorphic to $L_{n}$

We define $$L_{n}=\{A=(a_{ij})\in M_{n}(\mathbb{R})\mid \sum_{i=1}^{n} a_{ij}=0 \;\;\;\text{for every fixed j}\}$$
This is a Lie subalgebra of $M_{n}(\mathbb{R})$.
A dynamic-geometric proof for ...

**4**

votes

**0**answers

193 views

### Lie group structure over diffeomorphisms group

Let $M$ be a smooth manifold. Is it true that for every subgroup $G$ of $diff(M)$ there is at most one Lie group structure on $G$ such that the natural left $G$-action on $M$ is smooth?
Edit: by "Lie ...

**1**

vote

**1**answer

79 views

### Universal Chevalley group associated to $D_l$

Consider the simple Lie algebra $D_l$. Consider the universal Chevalley group $G$ over a field $K$ associated to it. Then $G$ is a subgroup of the orthogonal group $O_{2l}(K, f)$ where $f$ is the ...

**4**

votes

**1**answer

108 views

### Hua Luogeng's definition of automorphism group for Hermitian symmetric space

I'm trying to make sense of a definition appearing in Hua Luogeng's book "Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains".
Consider the Hermitian symmetric space ...

**1**

vote

**0**answers

92 views

### A question about the associative classes of parabolic subgroups

Let $\mathbb{A}$ denote the adele group of $\mathbb{Q}$. Suppose that $P_1$ and $P_2$ are parabolic subgroups of a reductive algebraic group $G$, and consider their Langlands decomposition
$$
P(\...