Example of a continuous affine group action

Question

Let $G$ be any locally compact group and $H$ be a compact group.

We know that a map $F: G \rightarrow G$ is called affine if there exists some $\alpha \in G$ and an automorphism $\Lambda:G\rightarrow G$ such that $F=\alpha\Lambda$.

A continuous action of $H$ on $G$ is a continuous map $\pi: H\times G \rightarrow G$ given by $\pi(h, g) = h.g$ such that $e.g =g$ for all $g\in G$ and $(hk).g= h. (k.g)$ for all $h, k \in H$, $g\in G$.

Hence a continuous action $\pi$ of $H$ on $G$ is called a continuous affine action if for each $h\in H$, the map $g\mapsto (h.g): G \rightarrow G$ is affine.

My Question: Could you please give me an example of such a continuous affine action where atleast one of the maps $g\mapsto (h.g): G \rightarrow G$ is (of course) affine, but not an automorphism?

Thank you in advance, for your help.

Answer 1

Let $G=(\mathbb{R},+)$ and $H=C_2=\langle t\rangle$. Then we may define: $$t.x=1-x=\Lambda(x)+1,$$ for all $x\in \mathbb{R}$, where $\Lambda(x)=-x$ for all $x\in \mathbb{R}$.

Alternatively, you could take $G=H=S^1\subset \mathbb{C}^\times$. For $g\in G$ and $h\in H$ we can define $h.g=hg=h\Lambda(g)$, where $\Lambda$ is the identity.

For a third example, consider $G=SU(2)\times S^1$ and $H=S^1\subset \mathbb{C}^\times$. Let $\Lambda_h\colon G\to G$ denote conjugation by:$$\left(\begin{array}{cc}h&0\\0&\bar{h}\end{array}\right)$$ on the factor $SU(2)$, and the identity on the factor $S^1$.

Then we may define $$h.g=(1,h)\Lambda_h(g).$$

Answer 2

As a slight generalization to the second example of tkf, for any locally compact (Hausdorff) group $H$ one can consider the group $\operatorname{Aff}(H)=H\rtimes \operatorname{Aut}(H)$ of affine automorphisms of it. By definition, any element of $\operatorname{Aff}(H)$ is of the form $(g,\phi)$, where $g\in H$ and $\phi:H\to H$ is a topological group automorphism. A $(g,\phi)\in \operatorname{Aff}(H)$ acts on $H$ like so:

$$(g,\phi):H\to H, x\mapsto g\phi(x)=l_g\circ \phi(x),$$

where $l_g$ is the left translation by $g$. Note that the group structure of $\operatorname{Aff}(H)$ is given by: $(g,\phi)(h,\psi)=(g\phi(h), \phi\circ \psi)$. In other words, what you call "affine maps" of $H$ assemble into a group that acts via affine maps on $H$.

In particular, if $H$ is a compact group, $G=\operatorname{Aff}(H)$ is a locally compact group, and $G\curvearrowright H$ is an action by affine automorphisms where most elements are not automorphisms (i.e. most elements don't fix the identity element $e_H$ of $H$). One can also take subgroups of $\operatorname{Aff}(H)$ (that has non-trivial translation part) to get further examples (in fact, any (faithful) affine action on $H$ is of this latter form, by definition).

For example if $H=\mathbb{T}^d$ ($d\in\mathbb{Z}_{\geq1}$), then $G=\operatorname{Aff}(\mathbb{T}^d)= \mathbb{T}^d\rtimes \operatorname{Aut}(\mathbb{T}^d)\cong \mathbb{T}^d\rtimes \operatorname{GL}(d,\mathbb{Z})$ (e.g. by https://math.stackexchange.com/a/4180086/169085). Taking the subgroup $\mathbb{T}^d\rtimes 1\leq G$, for $d=1$ we recover tkf's second example.

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Above I skipped one detail: since the action $\operatorname{Aff}(H)\curvearrowright H$ is supposed to be topological, we need to specify a topology on $\operatorname{Aff}(H)$. $\operatorname{Aff}(H)$ is a semidirect product, so as a set it is a cartesian product. Accordingly we may define the topology of $\operatorname{Aff}(H)$ as the product topology (see e.g. How to construct a topological group from a semidirect product of two subgroups). So specifying the topology of $\operatorname{Aff}(H)$ reduces to specifying a topology on $\operatorname{Aut}(H)$. The standard way of doing this is to consider $\operatorname{Aut}(H)$ as a subgroup of the group $\operatorname{Homeo}(H)$ of homeomorphisms of $H$, for which there is a variety of natural topologies, e.g. the compact-open topology (= topology of convergence on compact subsets) (see https://mathoverflow.net/q/413361/66883). In the example of the affine actions on the torus there are more options as there is a differentiable structure, and one can consider $\operatorname{Aut}(\mathbb{T}^d)$ as a subgroup of the group $\operatorname{Diff}^r(\mathbb{T}^d)$ of $C^r$ diffeomorphisms of the torus (or the group of volume preserving diffeomorphisms, ...). In any event, the continuity of the affine action will be pointwise continuity, which is straightforward to guarantee.