# Tagged Questions

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### How to show that $GL_n/U$ is birationally isomorphic to $B^-$?

It is said that $GL_n/U$ is birationally isomorphic to $B^-$. Here $U$ acts by right multiplication on $GL_n$. I think that $GL_n/U$ consisting of cosets. Two matrices in the same coset if any two ...
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### quotient by a group that acts almost freely

How can I show that if a compact lie group G acts almost freely and smoothly on a manifold M, then M/G is Hausdorff? (an action is almost free if $G_x$ is finite for all x $\in$ M)
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### Riemannian symmetric pair $(G,H)$ with H non-compact

Let $G$ denote a connected Lie group and $H$ a closed subgroup. Suppose that $\sigma$ is an involutive automorphism of $G$. Assume that $(G,H,\sigma)$ is a Riemannian symmetric pair. So far I have ...
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### How to describe $G/U$?

Let $G=SL_2(\mathbb{C})$ and let $U = \{\left( \begin{matrix} 1 & x \\ 0 & 1 \end{matrix} \right): x \in \mathbb{C}\}$. We have an action of $U$ on $G$ by right multiplication. By definition, ...
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### The Weyl group of $E_6$ acting on embedded circles

I want to know the number of components of the normalizer of an arbitrary circle subgroup $S$ of (the compact real form of) the exceptional Lie group $E_6$. This number will always be $1$ or $2$. ...
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### Action of $SO_n$ on $\mathbb{S}^{n-1}$ induces fibre bundle.

Real compact Lie group $SO_n$ acts smoothly and transitively on $\mathbb{S}^{n-1} \subseteq \mathbb{R}^n$ with obvious action. Isotropy subgroup of each point in $\mathbb{S}^{n-1}$ is isomoprhic to ...
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### Lifting group actions to universal covers

Let $\tilde{G}$ be universal cover of a lie group $G$. Then we can easily lift any action of $G$ on a connected manifold $X$ to an action of $\tilde{G}$ on the universal cover $\tilde X$ of $X$. (cf. ...
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### How can I see the set $S_k(V):=\{A: \mathbb R^k\rightarrow V: A\ \textrm{is linear and}\ \textrm{ker}(A)=\{0\}\}$ as an homogeneous space..

how can I see the set $S_k(V):=\{A: \mathbb R^k\rightarrow V: A\ \textrm{is linear and}\ \textrm{ker}(A)=\{0\}\}$ as an homogeneous space? Thanks..
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### Smooth group actions ----> “Action Lie groupoids”?

It is well known that any group action of a group G on a set X gives rise to the corresponding action groupoids, see http://math.ucr.edu/home/baez/week249.html , for instance. Now in a perfect ...
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### action of $O(n,\mathbb{R})$ on ${S}^{n-1}$

Is the action of $O(n,\mathbb{R})$ on ${S}^{n-1}$ transitive? I think this is true as orthogonal matrices are supposed to rotate and keep the length fixed, but how do I prove this? EDIT: Based on ...
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### The Quaternions and $SO(4)$

I am interested in the map $\phi:S^3 \times S^3 \to GL_4(\mathbb{R})$ given as follows: Let $(p,q) \in S^3 \times S^3$. We identify $p$ and $q$ as real quaternions with unit norms and define ...
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### Exponential intertwining of linear and local actions.

I am reading Duistermaat's 1973 paper on relating the convexity of the image of a moment map to the image of the fixed points of an antisymplectic involution. In that paper, the following comment is ...
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### Characteristic functions of group-invariant probability distributions

Suppose that we have a probability distribution $\rho(\mathbf x)$ on a manifold $\mathcal M$, which is invariant under the action of a Lie group $G$, $\rho(g\mathbf x)=\rho(\mathbf x)$ for all ...
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### Two definitions of equivariant sheaves

Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$. (a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action ...
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### Ergodic action of a group

What does it mean and how is it defined if the action of a group is meant to be ergodic? Thank you for your replies!
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### Action of a Lie group on a coset of its subgroup

I am a physicist, so sorry for the lack of rigor. It is well known that a (say compact) Lie group $G$ acts naturally by left multiplication on the coset space $G/H$ where $H\subset G$ is its (Lie) ...
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### Complexifying a group action of SL(n, R) to a group action of SL(n, C)

Given an analytic group action of $SL(n, \mathbb{R})$ on $\mathbb{R}^m$ fixing the origin, in this article the author then proceeds to "complexify the analytic $SL(n, \mathbb{R})$ action to obtain a ...
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### Finiteness of fixed points of a Lie group action

Let $\psi: G\rightarrow \mathrm{Diff}(M)$ be a smooth non-trivial action of a compact connected Lie group $G$ on a compact connected smooth manifold $M$. Under which assumptions there will be a ...
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### Infinitesimal generators of actions

Is there a method to obtain an action of an infinite dimensional Lie group starting with its infinitesimal generator ? I'm interested about actions of G on itself . And I was wondering if I can ...
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### What is the Lobachevsky space?

I am reading about Lie group actions on manifolds and the author used the Lobachevsky space $SO^{+}_{1,n}/SO_n$ with the actions of the subgroups $SO_n$, $SO_{1,n-1}$ and the horispherical group ...
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### A K-invariant submanifold of G-manifold and fundamental vector fields

Let a (connected) Lie group $G$ act on $M$. Assume that the action is locally free. (In other words, if the fundamental vector field of $X \in \mathrm{Lie(G)}$  \underline{X}(p) := ...
### If $\exp(itH) A \exp(-itH) = A$ for all $t$, do $A$ and $H$ commute?
Let $H$ be a self-adjoint $n \times n$ matrix with complex entries. $H$ gives rise to a continuous 1-parameter group of unitaries $t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n)$. Let $A$ be some ...
Let $M$ be a manifold (which you can assume compact if it helps) and consider the natural action of the isometry group of $M$, Iso($M$) on $M$. The we can define the symmetry rank of $M$ as the rank ...