Questions about Haar integral for the group $GL_2(\mathbb{R})$.

Question

I have some questions about Haar integral for the group $GL_2(\mathbb{R})$. How to show that a Haar integral for the group $GL_2 (\mathbb{R})$ is given by \begin{align} I(f ) & = \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} f\left( \begin{matrix} x & y \\ z & w \end{matrix} \right) \frac{dx dy dz dw}{|xw-yz|^2}? \end{align} I think that for $s=(x1, y1; z1, w1)$ we have \begin{align} I(L_sf ) & = \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} f ( \left( \begin{matrix} x1 & y1 \\ z1 & w1 \end{matrix} \right)^{-1} \left( \begin{matrix} x & y \\ z & w \end{matrix} \right)) \frac{dx dy dz dw}{|xw-yz|^2} \\ & = \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} f ( \frac{1}{|x1w1-y1z1|} \left( \begin{matrix} w1 & -y1 \\ -z1 & x1 \end{matrix} \right) \left( \begin{matrix} x & y \\ z & w \end{matrix} \right)) \frac{dx dy dz dw}{|xw-yz|^2}. \qquad (1) \end{align} How could we prove that (1) equals $$ \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} f ( \left( \begin{matrix} x & y \\ z & w \end{matrix} \right)) \frac{dx dy dz dw}{|xw-yz|^2}? $$ Thank you very much.

Answer 1

We only need to show that for any $s\in GL_2(\mathbb{R})$, $$\int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} f(s(\begin{matrix} x & y \\ z & w \end{matrix})) \frac{dxdydwdz}{|xw-yz|^2} = \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} f((\begin{matrix} x & y \\ z & w \end{matrix})) \frac{dxdydwdz}{|xw-yz|^2}. $$

We need the change of variables formula: $\int_{T(U)} f dx = \int_U f \circ T |J_T|dx$, where $U \subseteq \mathbb{R}^4$ is open, $T$ is an injective differentiable function, and $|J_T|$ is the Jacobian of $T$.

Let $$s=\left(\begin{matrix} x_1 & y_1 \\ z_1 & w_1 \end{matrix}\right).$$ Define a map: $T:GL_2 \to GL_2$, $$ T(\left(\begin{matrix} x & y \\ z & w \end{matrix}\right))= s \left(\begin{matrix} x & y \\ z & w \end{matrix}\right) = \left(\begin{matrix} x_1 & y_1 \\ z_1 & w_1 \end{matrix}\right) \left(\begin{matrix} x & y \\ z & w \end{matrix}\right). $$ The map $T$ sends $x \mapsto x_1 x + y_1 z$, $y \mapsto x_1 y + y_1 w$, $z \mapsto z_1 x +w_1 z$, $w \mapsto z_1 y + w_1 w$. Therefore the Jacobian $|J_T|$ of $T$ is $|x_1 w_1 - y_1 z_1 |^2$.

Let $F(x, y, z, w)=f((x, y; z, w))/|xw - yz|^2$. Then $$ F(x_1 x + y_1 z, x_1 y + y_1 w, z_1 x + w_1 z, z_1 y + w_1 w) = f(s(x, y; z, w))/(|xw - yz|^2|x_1w_1 - y_1z_1|^2) $$ and \begin{align} & \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} f(s \left(\begin{matrix} x & y \\ z & w \end{matrix}\right)) dxdydzdw/|xw - yz|^2 \\ & = \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} F(x_1 x + y_1 z, x_1 y + y_1 w, z_1 x + w_1 z, z_1 y + w_1 w) |J_T| dxdydzdw \\ & = \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} F \circ T \left(\begin{matrix} x & y \\ z & w \end{matrix}\right) |J_T| dxdydzdw \\ & = \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} F(\left(\begin{matrix} x & y \\ z & w \end{matrix}\right)) dxdydzdw \\ & = \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} \int_{\mathbb{R}} f(\left(\begin{matrix} x & y \\ z & w \end{matrix}\right)) dxdydzdw/|xw-yz|^2. \end{align}