Classify the conjugacy classes in \(\SL_2(\RR)\).
Solution
Let \(A\in\SL_2(\RR)\) and set \[ t=\operatorname{tr}(A). \] Its characteristic polynomial is \[ x^2-tx+1. \] The conjugacy classification is determined by the sign of \(t^2-4\), with an additional orientation sign in the elliptic and nontrivial parabolic cases.
1. Hyperbolic classes: \(|t|>2\).
The eigenvalues are distinct real numbers \(\lambda,\lambda^{-1}\). Thus \(A\) is \(\SL_2(\RR)\)-conjugate to \[ \begin{pmatrix}\lambda&0\\0&\lambda^{-1}\end{pmatrix}, \qquad \lambda+\lambda^{-1}=t. \] Swapping the two eigenvalues is achieved by \[ \begin{pmatrix}0&-1\\1&0\end{pmatrix}\in\SL_2(\RR), \] so there is one conjugacy class for each trace \(|t|>2\).
2. Elliptic classes: \(|t|<2\).
Write \[ t=2\cos\theta, \qquad 0<\theta<\pi. \] Then \(A\) is conjugate in \(\GL_2(\RR)\) to a rotation. In \(\SL_2(\RR)\) there are two classes: \[ R_\theta= \begin{pmatrix} \cos\theta&-\sin\theta\\ \sin\theta&\cos\theta \end{pmatrix}, \qquad R_{-\theta}. \] They are not \(\SL_2(\RR)\)-conjugate. Indeed, if \(P R_\theta P^{-1}=R_{-\theta}\), then, after subtracting \(\cos\theta I\) and dividing by \(\sin\theta\), one gets \[ PJP^{-1}=-J, \qquad J=\begin{pmatrix}0&-1\\1&0\end{pmatrix}. \] Every real matrix satisfying \(PJ=-JP\) has negative determinant unless it is singular, so no such \(P\) lies in \(\SL_2(\RR)\).
3. Parabolic classes: \(t=2\).
Either \(A=I\), or \(A\) is nontrivial unipotent. Every nontrivial unipotent is \(\SL_2(\RR)\)-conjugate to exactly one of \[ U_+=\begin{pmatrix}1&1\\0&1\end{pmatrix}, \qquad U_-=\begin{pmatrix}1&-1\\0&1\end{pmatrix}. \] They are not \(\SL_2(\RR)\)-conjugate: a \(\GL_2(\RR)\)-conjugator between them has negative determinant, while the centralizer of \(U_+\) consists of matrices \[ \begin{pmatrix}a&b\\0&a\end{pmatrix} \] with determinant \(a^2>0\). Hence the sign cannot be changed by a determinant-\(1\) conjugator.
4. Parabolic classes: \(t=-2\).
Likewise there are exactly three classes: \[ -I, \qquad -U_+, \qquad -U_-. \]
Therefore trace determines the hyperbolic class, while for each elliptic trace there are two classes and for each trace \(\pm2\) there are the scalar class and two nontrivial parabolic classes.