What does $H=GL(2,\mathbb{R})/(Z(GL(2,\mathbb{R}))\cdot O(2,\mathbb{R}))$ suggest?

Allow $H=\left\{ z\in\mathbb{C}\mid\Im\left(z\right)>0\right\}$ be the upper - fifty percent Poincare aircraft. Allow $GL\left(2,\mathbb{R}\right)$ be the basic straight team, $Z\left(GL\left(2,\mathbb{R}\right)\right)$ be the facility of the basic straight team and also $O\left(2,\mathbb{R}\right)$ be the orthogonal subgroup of $GL\left(2,\mathbb{R}\right)$.

What does it suggest to claim $H=GL\left(2,\mathbb{R}\right)/\left(Z\left(GL\left(2,\mathbb{R}\right)\right)\cdot O\left(2,\mathbb{R}\right)\right)$? The left - hand side is a statistics room and also the right-hand man side is a set of cosets of $GL\left(2,\mathbb{R}\right)$. So I'm perplexed concerning what it suggests to write that they are equivalent or to claim "the upper fifty percent aircraft is ..." It feels like this would certainly be the team of alignment preserving isometries of H, yet I still locate the terms perplexing.

I've been attempting to identify what this can perhaps suggest, yet my searches on the net have actually not been rewarding. I've additionally considered 2 resources on typical modular teams yet they make no reference of this reality. A description or reference would certainly be substantially valued.

Inspiration : I read a paper labelled "On Modular Functions in particular p" by Wen - Ch' ing Winnie Li which can be located at http://www.jstor.org/stable/1997973. The case shows up on web page 3 of the pdf (web page 232 of the journal). It is additionally mentioned on the wikipedia web page : http://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model

2019-12-02 02:53:58
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Answers: 2

H is not a team, yet instead a room with a team acting upon it.

If K ≤ G is a subgroup, after that G/K is not generally a team, yet instead a set of cosets on which G acts. G/K is just a team if K is regular.

In your scenario, the subgroup is not regular, so you just get a team activity.

One keyword kind the wikipedia write-up is "isotropy subgroup". If G acts transitively on a room H, after that the set K of components of G that deal with some certain factor of H creates a subgroup (the "isotropy subgroup"), and also the activity of G on H is isomorphic to the activity of G on the cosets in G/K.

2019-12-03 04:24:39

My remarks were obtaining also long, so I will certainly upload this as a solution : the isomorphism described is an isomorphism of $G$ - collections. This operates in wonderful generalization : allow $H$ be a set, allow $G$ act upon $H$ transitively. Select an approximate factor $z\in H$ and also allow $K=\text{Stab}_G(z)$ be the factor stabiliser in $G$. Keep in mind that $K$ is not regular as a whole, given that the team $gKg^{-1}$ secures the factor $g(z)$ (my activity gets on the left). So, $K$ is regular if and also just if it acts trivially on $H$.

However, the set of cosets $G/K$ is constantly a $G$ - set, i.e. a set with an activity of $G$ : $$g: hK\mapsto (gh)K$$ for all $g\in G$ and also $hK\in G/K$. This is the common coset activity. Currently, examine that the map $$\phi:G/K\rightarrow H,\;gK\mapsto g(z)$$ is a bijection of $G$ - collections, i.e. a bijection of collections that values the $G$ - activity.

2019-12-03 04:22:43