Representation theory of SL2(R)

Example

Let $G = \mathrm{SL}_2(\R)$, the group of real $2 \times 2$ matrices of determinant $1$. It is a connected simple Lie group that is not compact. So each nontrivial irreducible unitary representation of $G$ is infinite-dimensional. Indeed, a nontrivial unitary representation on $\C^n$ has a differential that embeds $\mathfrak{sl}_2(\R)$ into $\mathfrak{u}(n)$. Each subalgebra of $\mathfrak{u}(n)$ has an inner product for which every $\operatorname{ad} X$ is skew-symmetric. The algebra $\mathfrak{sl}_2(\R)$ has none, since $\operatorname{ad} X$ has real nonzero eigenvalues for $X = \operatorname{diag}(1, -1)$.

Principal series. Fix $\varepsilon \in \set{0, 1}$ and $\nu \in i\R$. Let $H_{\varepsilon, \nu}$ be the space of functions $f$ on $\R^2 \setminus \set{0}$ with $f(\lambda x) = \sgn(\lambda)^{\varepsilon} \abs{\lambda}^{-1-\nu} f(x)$ for $\lambda \ne 0$. Its norm is the norm of $L^2$ on the unit circle. The action is $(\pi(g) f)(x) = f(g^{-1} x)$. Here $\abs{f}^2$ is homogeneous of degree $-2$. For such functions the integral over the circle is unchanged by linear maps of determinant $1$, so $\pi$ is unitary.

Discrete series. For an integer $k \ge 2$, let $D_k^+$ be the space of holomorphic functions $f$ on the upper half plane $\set{z = x + iy : y > 0}$ with $\int \abs{f(z)}^2 y^{k-2} \, dx \, dy < \infty$. For $g^{-1} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$, the action is $(\pi(g) f)(z) = (cz + d)^{-k} f\big(\frac{az + b}{cz + d}\big)$. This is an irreducible unitary representation whose matrix coefficients are square integrable on $G$.