---
tag: 5E547TJ7PS
kind: example
name: Representation theory of SL2(R)
msc: ["22E46", "22E45"]
illustrates: [7JA7WH92K0]
uses: [Q0GBCB9RA4, 7JA7WH92K0, 74BPYJNZTM, FX3NEFYFFN]
sources:
  - url: https://en.wikipedia.org/wiki/Representation_theory_of_SL2(R)
same_as:
  wikidata: Q7314223
  wikipedia: Representation theory of SL2(R)
---

## Example

Let $G = \mathrm{SL}_2(\R)$, the [[group|Q0GBCB9RA4]] of real $2 \times 2$ matrices of determinant $1$.
It is a connected simple Lie group that is not compact.
So each nontrivial [[irreducible|FX3NEFYFFN]] [[unitary representation|74BPYJNZTM]] 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 $G$.

## Remarks

Bargmann classified the irreducible unitary representations of $G$ in 1947.
Up to equivalence they are the following.

1. The trivial representation.
2. The irreducible principal series.
3. The complementary series, a family with a real parameter in an interval.
4. The discrete series $D_k^{\pm}$ for $k \ge 2$.
5. The two limits of discrete series $D_1^{\pm}$, the two pieces of the reducible principal series with $\varepsilon = 1$ and $\nu = 0$.

Here $D_k^-$ is the complex conjugate of $D_k^+$, realized on antiholomorphic functions.

The Casimir element of $\mathfrak{sl}_2(\R)$ acts on each irreducible representation by a scalar, and this scalar is a main invariant in the classification.
These representations govern the spectral theory of the Laplacian on hyperbolic surfaces $\Gamma \backslash \mathbb{H}$, and holomorphic cusp forms of weight $k$ give copies of the discrete series $D_k^{\pm}$ in $L^2(\Gamma \backslash G)$.

## Explanations

### L1 human

The group $\mathrm{SL}_2(\R)$ moves the points of the hyperbolic plane.
Unlike a compact group, it has no nontrivial length-preserving action on a finite-dimensional space.
So its interesting representations live on spaces of functions, such as functions on the circle or holomorphic functions on the upper half plane.
Bargmann found all of them, and they come in a few families.