Questions about Haar integral.

Question

Questions about Haar integral. Let $B$ be the subgroup of $GL_2 (\mathbb{R})$ defined as $$ B =\{ \left( \begin{matrix} 1 & b \\ 0 & c \end{matrix} \right), b, c \in \mathbb{R}, c \neq 0 \}. $$ How to show that $$ I(f ) = \int_{\mathbb{R}^\times } \int_{\mathbb{R}} f\left( \begin{matrix} 1 & b \\ 0 & c \end{matrix} \right) db \frac{dc}{c} $$ is a Haar integral on $B$. I think that for $s = (1, b1; 0, c1)$, we have $$ I(L_sf ) \\ = \int_{\mathbb{R}^\times } \int_{\mathbb{R}} f(s^{-1} \left( \begin{matrix} 1 & b \\ 0 & c \end{matrix} \right)) db \frac{dc}{c} \\ \int_{\mathbb{R}^\times } \int_{\mathbb{R}} f\left( \begin{matrix} 1 & b-\frac{cb1}{c1} \\ 0 & \frac{c}{c1} \end{matrix} \right) db \frac{dc}{c}. \qquad (1) $$ But (1) is not equal to $$ \int_{\mathbb{R}^\times } \int_{\mathbb{R}} f\left( \begin{matrix} 1 & b-\frac{cb1}{c1} \\ 0 & \frac{c}{c1} \end{matrix} \right) d(b-\frac{cb1}{c1} ) \frac{d(c/c1)}{c/c1}. $$ Thank you very much.

Answer 1

We only need to show that for any $s\in B$, $\int_{\mathbb{R}^{\times}} \int_{\mathbb{R}} f(s(\begin{matrix} 1 & b \\ 0 & c \end{matrix})) db dc/c = \int_{\mathbb{R}^{\times}} \int_{\mathbb{R}} f((\begin{matrix} 1 & b \\ 0 & c \end{matrix})) db dc/c $. We need the change of variables formula: $\int_{T(U)} f dx = \int_U f \circ T |J_T|dx$, where $U \subseteq \mathbb{R}^n$ is open, $T$ is an injective differentiable function, and $|J_T|$ is the Jacobian of $T$.

Let $s=(1, x; 0, y)$ The Jacobian of the map: $T:B \to B$, $T((1, b; 0, c))=s(1, b; 0, c)$ is $b \mapsto b+xc$, $c \mapsto cy$. Therefore the Jacobian $|J_T|$ of $T$ is $y$. Let $F(b, c)=f((1, b; 0, c))/c$. Then
$$\int_{\mathbb{R}^{\times}} \int_{\mathbb{R}} f(s(\begin{matrix} 1 & b \\ 0 & c \end{matrix})) db dc/c = \int_{\mathbb{R}^{\times}} \int_{\mathbb{R}} F(b,c) db dc \\ = \int_{\mathbb{R}^{\times}} \int_{\mathbb{R}} F \circ T (b,c) y db dc \\ = \int_{\mathbb{R}^{\times}} \int_{\mathbb{R}} F(b+cx,cy) y db dc \\ = \int_{\mathbb{R}^{\times}} \int_{\mathbb{R}} (f(s(1,b; 0,c))/(cy)) y db dc \\ = \int_{\mathbb{R}^{\times}} \int_{\mathbb{R}} f(s(1,b; 0,c))db dc/c. $$