How to find coordinates on the quotient

Question

I need to work on the quotient space $SL(2,\mathbb R)/SO(2,\mathbb R)$ and I am having trouble finding coordinates on this space.

Any help would be really appreciated.

Answer 1

Let us set

$$Q=SL(2,\mathbb R)/SO(2,\mathbb R).$$

A parameterization of $Q$ can be obtained by using Iwasawa decomposition of $SL(2,\mathbb{R})$ as described in this excellent document:

$$\tag{1}S=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\ \ \cos\theta\end{pmatrix}\begin{pmatrix}r&0\\ 0&1/r\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}$$

Thus, an element of $SL(2,\mathbb{R})$ is determined by 3 parameters:

$$\tag{2}\theta \in [0, 2 \pi), r \in \mathbb{R}_+^*, x \in \mathbb{R}.$$

Roughly said, an element of $Q$ is obtained by forgetting parameter $\theta$, making $Q$ isomorphic to the set of matrices of the form:

$$\begin{pmatrix}r&0\\ 0&1/r\end{pmatrix}\begin{pmatrix}1&x\\ 0&1\end{pmatrix}$$

which, moreover, constitutes a group parameterized by $r$ and $x$.

We shouldn't say too hastily that this provides a "good" parametrization of $Q$ because the group of matrices defined by (2) is not a normal subgroup of the group defined by (1) (see document).

Another interesting aspect given in this document is the homeomorphy of $SL(2,\mathbb{R})$ to a torus, giving an homeomorphy of $Q$ to a disk.

Another enlighting document is this one.

A very complete reference for $SL(2,\mathbb{R})$ is the book plainly entitled ... $SL_2(\mathbb{R})$ by Serge Lang.

See also "The Heat Kernel and Theta Inversion on $SL_2(\mathbb{C})$" by Jay Jorgenson and the same Serge Lang.

Have also a look at the Iwasawa decomposition page on Wikipedia.

Answer 2

$SL(2,\mathbb{R})$ acts on the space $V$ of real symmetric 2 by 2 matrices by:

$g.x = gxg^T$

The stabilizer of $I_2$ is $SO(2)$, so that the quotient space $SL(2,\mathbb{R})/SO(2)$ is nothing but the orbit of $I_2$ under this action. Note that the determinant is invariant under this action. Moreover, any element in the orbit is positive definite. Conversely, let $x$ be symmetric positive definite matrix of determinant $1$. By using a basis consisting of an oriented orthonormal basis with respect to the inner product on $\mathbb{R}^2$ defined by $x$, one can see that any such $x$ is in the orbit of $I_2$.

Hence, the quotient space can be described as the space

$\{(a,b,c) \in \mathbb{R}^3; a>0, c>0 \text{ and } ac-b^2 = 1 \}$

It can also be thought of as the space of real quadratic polynomials in one real variable which are positive everywhere and have a fixed negative value of the discriminant. Finding now local coordinates is straightforward.

Edit: by one more change of variable, the quotient space can also be thought of as the hyperbolic plane. There is also a more direct way of viewing the quotient space as the upper half plane, namely using Moebius transformations.