# Examples of group $G=N \rtimes H$ where $N$ and $H$ are as below

I am searching for examples of connected locally compact group $$G = N \rtimes H$$, where $$N$$ is a simply connected nilpotent non-abelian Lie group, $$H$$ is linear reductive and $$H$$ operates on $$N$$ without non-trivial fixed points. Please enlighten me.

P.S. I added the ergodic theory tag because I believe such groups are seen there.

This is also posted in stackexchange. https://math.stackexchange.com/questions/3037426/examples-of-non-abelian-simply-connected-nilpotent-lie-groups

• I would try to stick $H$ into the title somehow, because without it the question seems really trivial (all groups of upper triangular matrices are contractible) Dec 13, 2018 at 8:18

Consider the nilpotent group $$N={\mathbb R} \rtimes {\mathbb R}^2$$ (the group of $$3\times 3$$ upper triangular unipotent matrices with real coefficients. If $$v,w \in {\mathbb R}^2$$, then their commutator $$[v,w]$$ in $$N$$ is simply the wedge $$v\wedge w \in \wedge ^2 {\mathbb R}^2\simeq {\mathbb R}$$.

The group $$H=SL(2,{\mathbb R})$$ operates on $$N$$ since it preserves the symplectic form $$v\wedge w$$ on $${\mathbb R}^2$$.

• Is it obvious that $H$ acts without fixed points? Dec 13, 2018 at 8:20
• If the action of $H$ on $N$ is as follows: for $h \in H, (t,v) \in N, h.(t,v) = (t,hv)$ then $H$ fixes $\mathbb{R} \rtimes \{(0,0)\}$ ! What exactly is your action? Dec 13, 2018 at 8:37
• OK. You replace $SL(2,{\mathbb R})$ by $H=GL(2,{\mathbb R})$ except that on $\mathbb R$ the group $H$ acts via determinant, and hence does not fix anything except identity. It acts on ${\mathbb R}^2$ preserving the symplectic form up to sclars, and hence still acts on $N$. Dec 13, 2018 at 11:54
• $N$ is not a semidirect product $\mathbf{R}\rtimes\mathbf{R}^2$; I don't know if you mean to think of it as a central (not split) extension with kernel $\mathbf{R}$ and quotient $\mathbf{R}^2$, or as a semidirect product with kernel $\mathbf{R}^2$ and quotient $\mathbf{R}$. Anyway, it works.
– YCor
Dec 13, 2018 at 19:33
• @Mambo: yes. The simple connectedness follows from the homotopy exact sequence since $N \rightarrow {\mathbb R}^2$ is a locally trivial fibration with fibre $\mathbb R$. Dec 15, 2018 at 8:09

You can take $$H=T$$ to be the maximal torus in a (quasi)split reductive group $$\Gamma$$, $$N$$ to be the unipotent radical of a Borel subgroup containing $$T$$, and the action is by conjugation. If some element $$x\in N$$ is fixed by $$T$$ then it centralizes $$T$$ and so must be in $$T$$, which is impossible.

• You need to exclude the edge case that $N$ is abelian (e.g., $SL_n$ works for $n \geq 3$). I think more generally $H$ can be taken to be a Levi factor of a parabolic subgroup $P=L \ltimes U$ with the same caveat (your example is the case $P=B = T\ltimes N$). Dec 14, 2018 at 17:45