# Questions tagged [lie-groups]

Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.

1,918 questions

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### History of the notion of $(G,X)$-structure

I'm currently searching for sources and historical basis on the notion of $(G,X)$-structure as it appears in Thurston's work.
So far, it appears that he was the first to set it. Many mathematicans ...

**0**

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**0**answers

49 views

### Lattices are not solvable in non-compact semisimple Lie groups

I'm trying to prove the following result.
If $G$ is a non compact semisimple Lie group with no compact factors (lying in some $SL(l,\mathbb{R})$), and $\Gamma$ is a lattice in $G$, then $\Gamma$ is ...

**3**

votes

**1**answer

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### Computing Deligne-Lusztig Characters in General

The goal for this question is to try to find a relatively explicit way of computing the Deligne-Lusztig characters. I understand that the $R_{T,\theta}$ can be computed if we know the values of the ...

**2**

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39 views

### Generalizing polynomial identities for rings

For a ring $R$, a polynomial identity of $R$ is a polynomial (in non-commuting variables) $f(x_1,\ldots,x_n)\in \mathbb{Z}[x_1,\ldots, x_n]$ such that for any choice of $a_i\in R$, $f(a_1,\ldots, a_n)=...

**7**

votes

**1**answer

211 views

### Independence of Duistermaat-Heckman measure

Suppose that a compact Kähler manifold $(X,\omega)$ has a real torus acting on it by symplectomorphisms in a Hamiltonian way (the torus is not necessarily of maximal rank). Then for any smooth ...

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27 views

### Reference Request: Carnot Groups over Complexes

Is there a theory of complex (analytic) Carnot groups and Caratheodory metrics?

**2**

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**1**answer

119 views

### On a criterion for rational-smoothness of Schubert varieties and an ambiguity of the taking the ambient Algebraic group to be simply connected or not

In the paper: Pattern Avoidance and Rational Smoothness of
Schubert Varieties, Sara C. Billey, Advances in Mathematics 139, 141-156(1998), https://www.sciencedirect.com/science/article/pii/...

**11**

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454 views

### Nonnegativity of an integral over the unitary group

For an $n$-by-$n$ unitary matrix $U$ and a permutation $\sigma\in S_n$, let
$$w_\sigma=(-1)^\sigma\det(U^*)\prod_{i=1}^n U_{i,\sigma(i)}.$$
Is $\int_{U(n)}\mathrm{Re}(w_{\sigma_1})\mathrm{Re}(w_{\...

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46 views

### Uniqueness of Equivariant Harmonic Map for Surface Group Representation

In section 1.2 of https://arxiv.org/pdf/1311.2919.pdf the following result is stated.
$\textbf{Theorem}$ (Labourie). Let $S$ be a closed Riemann surface of negative Euler characteristic, $Γ$ its ...

**0**

votes

**1**answer

159 views

### On some notations and notions of a paper on smoothness of Schubert varieties by Lakshmibai and Sandhya

I am reading the paper Criterion for smoothness of Schubert varieties in $\mathrm{Sl}(n)/B$ by V Lakshmibai and B Sandhya; Proc. Indian Acad. Sci. (Math. Sci.), Vol. 100, No. 1, April 1990, pp. 45-52. ...

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72 views

### Peter–Weyl theory for vector fields

Let $G$ be a compact Lie group. The classical Peter-Weyl theorem shows that $L^2(G)$ splits into an orthogonal direct sum of irreducible finite-dimensional unitary representations of $G$. This is a ...

**3**

votes

**1**answer

65 views

### Convex Hulls of Demazure Modules

Let $G$ be a semisimple algebraic group over $\mathbb{C}$ and for a highest weight $\lambda$, denote by $V_{\lambda}^w$ the Demazure module associated with $\lambda$ and $w$. More precisely, $V_{\...

**2**

votes

**1**answer

67 views

### Exhaustion of restrictions of holomorphic / antiholomorphic representations

Let $G$ be a simple Lie group of Hermitian type, and $G'$ be a reductive subgroup of $G$. Suppose that $G'$ is also of Hermitian type and contains the center of the maximal compact subgroup of $G$. ...

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**1**answer

85 views

### When are Carnot groups negatively curved and homeomorphic to Euclidean space

When are Carnot groups complete and negatively curved (in the sense of $CAT(\kappa)$ spaces)?

**5**

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**1**answer

229 views

### Automorphism group of the special unitary group $SU(N)$

Let us consider the automorphism group of the special unitary group $G=SU(N)$.
We know there is an exact sequence:
$$
0 \to \text{Inn}(G) \to \text{Aut}(G) \to \text{Out}(G) \to 0.
$$
For $G=SU(2)...

**3**

votes

**0**answers

50 views

### Transformation between nearby tangent planes [closed]

This question is kinda long, but the picture is quite clear.
Question: Let $(M,g)$ be a Riemannian manifold, $p$ a point on $M$, $U$ an open neighborhood of $0\in T_pM$ such that $exp_p|_U$ is a ...

**3**

votes

**2**answers

226 views

### How many non-isomorphic extensions with kernel $S^1$ and quotient cyclic of order $p$?

I want to determine how many non-isomorphic extensions (as group they are non-isomorphic) are possible of the form $1 \to \mathbb{S}^1 \to G \to (\mathbb{Z}_p)^k \to 1$, where $G$ is a compact lie ...

**1**

vote

**1**answer

67 views

### Doubling constant of Carnot group

This post shows that every Carnot group is a doubling metric space. However, what is its doubling constant?

**2**

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**1**answer

81 views

### Nice Form of Vector Field

Let $G$ be a reductive algebraic group (maybe reductive is not necessary) over an algebraically closed field $k$ of characteristic zero. Let $X$ be a homogeneous affine $G$-variety, i.e. $X=G/K$ for ...

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147 views

### What are the points about representation of groups? [closed]

For a fixed (let say finite-, or Lie-, to respect the historical motivations) group, why does the study of all its linear representations over a fixed field, leads to some knowledge about its ...

**9**

votes

**1**answer

171 views

### Reference requests: Integral cohomology of $G_2$-homogeneous spaces

Do you know a place where the integral cohomology of $G_2$-homogeneous spaces is computed?
Great computational efforts using representation theory in order to determine the ...

**5**

votes

**1**answer

80 views

### What is the connection between Frechet Lie groups and Lie algebras?

An ordinary Lie group has a differentiable manifold structure, i.e. it is locally isomorphic to a finite-dimensional Euclidean space. A Frechet Lie group, on the other hand, has a Frechet manifold ...

**4**

votes

**2**answers

208 views

### Non-faithful irreducible representations of simple Lie groups

For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.
...

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137 views

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### Relation between groups $A_n$, $B_n$, $D_n$ and $S_n$ or inversions of random elements in Coxeter groups

First of of all I'm trying to find a general interpretation to the following facts below.
Let's look at the property of Kendall-Mann numbers $M(n)$ which are row maxima of Triangle of Mahonian ...

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votes

**1**answer

152 views

### Is Sp(1).Sp(1).Sp(1) the homotopy-fixed locus of Triality?

The group
$$
Sp(2)\cdot Sp(1) := \big( Sp(2) \times Sp(1)\big)/\big\{(1,1), (-1,1)\big\}
$$
is canonically a subgroup of $Spin(8)$ in three different ways, these representing three distinct ...

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71 views

### What is the difference between Sp(2)S^1 and Sp(2)?

I am reading B. Wilking's publication, which said that $Sp(2)S^1$ is the normalizer of $Sp(2)\subset SU(5)$. I consider that $Sp(2)$ can be embedded into $SU(5)$ by $A+jB\rightarrow$
$\left(\begin{...

**2**

votes

**1**answer

96 views

### Characterisation of even nilpotent elements in $\mathfrak{sl}_n$

Is there a ''nice'' classification of even nilpotent elements in $\mathfrak{sl}_n,$ using the correspondence between nilpotent elements and partitions of n? By an even element, I mean an element $e$, ...

**3**

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**1**answer

156 views

### Distinguished dominant integral weight related to a branching problem

Let $G$ be a simple compact connected Lie group and let $K$ be a connected closed subgroup of $G$.
Let $\widehat G$ and $\widehat K$ denote the corresponding unitary duals, that is, the (equivalence ...

**5**

votes

**1**answer

161 views

### “Dimension” of discrete subgroups of infinite covolume in Lie groups

Let $G$ be a semisimple Lie group with finite center, $K$ a maximal compact
subgroup, and $d=\dim(G/K)$. Let $\Gamma$ be a non-cocompact discrete subgroup of $G$. [Edit: assume that $\Gamma$ is ...

**3**

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113 views

### Existence of loxodromic elements in certain subsets of $\text{PSL}_2(\mathbb C)$

Let $R$ be a subset of $\text{PSL}_2(\mathbb C)$ and consider its natural action on $\mathbb {CP}^1$. We say that $R$ is elementary if either $R$ is conjugated to a subset of $\text{SU(2)}$ or if ...

**2**

votes

**1**answer

237 views

### Malcev's paper “On a class of homogeneous spaces” in English

I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...

**5**

votes

**1**answer

156 views

### Volume of balls in homogeneous manifolds

Let $X=G/H$ be a homogeneous manifold, where $G$ and $H$ are connected Lie groups and assume there is given a $G$-invariant Riemannian metric on $X$.
Let $B(R)$ be the closed ball of radius $R>0$ ...

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votes

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223 views

### Existence of an isotopy in Riemannian manifold

Let $(M,g)$ be a Riemannian manifold, and $p,q\in M$ be two fixed points. We assume $p,q$ are close enough. Say, we assume $p$ and $q$ are in the same normal coordinate chart. It is clear that there ...

**12**

votes

**3**answers

544 views

### Reference request: Grassmannian and Plucker coordinates in type B, C, D

Grassmannian $Gr(k,n)$ is the set of $k$-dimensional subspace of an $n$-dimensional vector space. What are the Grassmannian in types B, C, D? What are the analog of Plucker coordinates and Plucker ...

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votes

**0**answers

49 views

### Homogeneous space for intersection of subgroups

Suppose we have a Lie group $G$ and two subgroups $P_1$ and $P_2$. We can then study the homogeneous spaces $M_1=G/P_1$ and $M_2=G/P_2$, and bundles on these spaces associated to representations of $...

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votes

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210 views

### Gram-Schmidt map as a Riemannian submersion

We equip $\mathrm{GL}(n,\mathbb{R})$ and $\mathrm{O}(n)$ with their left-invariant metrics, whose restrictions to the corresponding neutral elements is the standard inner product $\mathrm{Trace}(AB^{\...

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**3**answers

190 views

### Real points of reductive groups and connected components

Let $\mathbf G$ be a connected reductive group over $\mathbb R$, and let $G = \mathbf G(\mathbb R)$. Then $G$ is not necessarily connected as a Lie group, e.g. $\mathbf G = \operatorname{GL}_n$. ...

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votes

**1**answer

140 views

### An easier reference than “On the Functional Equations Satisfied by Eisenstein Series”?

I'd like to learn about Eisenstein series so I started reading "On the Functional Equations Satisfied by Eisenstein Series"by Langlands.
http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/...

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vote

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77 views

### Basic notation question involving Lie Groups and Lie algebras

I just started reading "On the functional equations satisfied by Eisentstein series" by Langlands http://publications.ias.edu/sites/default/files/Eisenstein-ps.pdf . I wasn't sure of some notation/...

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### Compact image of adjoint action

Let $G$ be a connected Lie group and $H$ a connected Lie-subgroup.
Suppose that for every compact subset $K\subset G/H$ the $H$-orbit $H.K$ is relatively compact in $G/H$.
Is it true that the image ...

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votes

**3**answers

299 views

### $SO(m+1)$-equivariant maps from $S^m$ to $S^m$

Let $G=SO(m+1)$ , $m \geq 2$, act in the standard way on $S^m$.
Let $F:S^m \to S^m$ be a $G$-equivariant map, i.e., $g F(g^{-1}x) =F(x)$ for all $x \in S^m$ and $g \in G$.
Question 1: Is F the ...

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votes

**1**answer

115 views

### Simple modules for direct sum of simple Lie algebras

I think that the following statement is true, but I do not know how to prove it.
Let $\mathfrak{g}_1$ and $\mathfrak{g}_2$ be two real simple Lie algebras. If $M$ is a (infinite dimensional) complex ...

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votes

**1**answer

220 views

### Homotopy classes of maps between special unitary Lie groups

I am sorry to mislead the notations: $SU(n)$ should be replaced by $PSU(n)$. I will reformulate it now.
We consider the special unitary Lie group $SU(n)$. Then its center is $\mathbb{Z}_n$ and we ...

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vote

**1**answer

196 views

### Homotopy of group actions

Let $G$ be a topological group and $X$ be a topological space.
Let $\alpha$, $\beta:G\times X\to X$ be two group actions. We say that these two actions are homotopic actions if there is a continuous ...

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139 views

### More general form of Fourier inversion formula

My question begins as follows: Suppose $G$ is a compact (Lie) group and $A$ is a $G$-module, and denote the action of $G$ on $A$ by $\alpha$. Fix $a\in A$ and view
$$
f:g\mapsto \alpha_g(a)
$$
as an $...

**5**

votes

**1**answer

167 views

### Definition of a Dirac operator

So it seems that a Dirac operator acting on spinors on $\psi=\psi(\mathfrak{su}(2),\mathbb{C}^2)$ can be written in this case simply as:
$D=\sum_{i,j} E_{ij}\otimes e_{ji}$, where $E_{ij}$ are ...

**3**

votes

**1**answer

91 views

### A converse of Cartan's automatic continuity theorem

Let $G$ be a compact real Lie group. We say that $G$ has property $(*)$ if every abstract automorphism of $G$ is continuous. A theorem of Cartan says that if $G$ has perfect Lie algebra, it has ...

**2**

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**0**answers

118 views

### Milnor's conjecture on Lie group (co)homology and forgetful functor of extensions

Let $G$ and $H$ be compact Lie groups, Consider $Ext_{Lie}(G,H)$ the set of isomorphism of extensions of Lie groups:
$$
1\rightarrow G\rightarrow M\rightarrow H\rightarrow 1
$$
There exists a ...

**3**

votes

**1**answer

212 views

### Extensions of compact Lie groups

Let $G$, $H$ be two compact Lie groups (possibly disconnected). Two short exact sequences of compact Lie groups
$$
0\rightarrow G\rightarrow M_1 \rightarrow H\rightarrow 0,
$$
$$
0\rightarrow G\...

**5**

votes

**2**answers

239 views

### Compact connected Lie groups isomorphic as groups and manifolds

Let $G_1$ and $G_2$ be compact connected (not necessarily semi-simple) Lie groups. Assume that the underlying smooth manifolds of $G_1$ and $G_2$ are diffeomorphic and that the underlying abstract ...