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Let $\mathbf{M}_{1},\ldots,\mathbf{M}_{N}$ be invertible $2\times2$ matrices with entries in $\mathbb{Q}$ of the form:

$$\mathbf{M}_{n}=\left[\begin{array}{cc} a_{n} & b_{n}\\ 0 & c_{n} \end{array}\right],\textrm{ }\forall n\in\left\{ 1,\ldots,N\right\}$$ and let $G$ be the subgroup of $\textrm{GL}_{2}\left(\mathbb{Q}\right)$ generated by the $\mathbf{M}_{n}$s.

0) Is there a locally-compact abelian group topology on $G$ other than the discrete topology? Or must I roll with the discrete topology?

1) What is the Pontryagin dual, $\hat{G}$, of $G$? That is to say, what is it isomorphic to as an abstract group, and, given any $\chi\in\hat{G}$, what does the formula for: $$\chi\left(\left[\begin{array}{cc} a & b\\ 0 & c \end{array}\right]\right)$$ look like for an arbitrary $\left[\begin{array}{cc} a & b\\ 0 & c \end{array}\right]\in G$? More generally, what are the formulae for arbitrary continuous multiplicative characters $\chi:G\rightarrow\mathbb{C}\backslash\left\{ 0\right\}$?

2) Up to a multiplicative constant, what is a formula for a left-invariant differential form $d\mu$ on $G$ so that the associated left-haar measure $\mu$ of a given measurable subset $E\subseteq G$ can be written as: $$\mu\left(S\right)=\int_{S}d\mu$$ Or is $\mu$ just the counting measure?

3) When (if ever) is $G$ a unimodular group? For those cases where $G$ is not unimodular, what is the formula for a general right-invariant differential form on $G$?

4) Up to a multiplicative constant, what is a formula for a left-invariant (and/or right-invariant) differential form $d\nu$ on $\hat{G}$ so that the associated left-haar (and/or right-haar) measure $\nu$ of a given measurable subset $E\subseteq \hat{G}$ can be written as: $$\nu\left(S\right)=\int_{S}d\nu$$


Edit: On noting that: $$\chi\left(\mathbf{A}\mathbf{B}\right)=\chi\left(\mathbf{A}\right)\chi\left(\mathbf{B}\right)=\chi\left(\mathbf{B}\right)\chi\left(\mathbf{A}\right)=\chi\left(\mathbf{B}\mathbf{A}\right)$$

for all $\mathbf{A},\mathbf{B}\in G$ and all $\chi\in\hat{G}$, it then follows that every $\chi\in\hat{G}$ is uniquely determined by the values of $\chi\left(\mathbf{M}_{n}\right)$, for $n\in\left\{ 1,\ldots,N\right\}$. So does this mean an arbitrary continuous multiplicative character is: $$\chi\left(\mathbf{A}\right)=\prod_{n=1}^{N}z_{n}^{\left(\#\textrm{ of }\mathbf{M}_{n}\textrm{s in }\mathbf{A}\right)-\left(\#\textrm{ of }\mathbf{M}_{n}^{-1}\textrm{s in }\mathbf{A}\right)}$$ where the $z_{n}$s are any $N$ non-zero complex numbers, or do I need to take the determinants of the $\mathbf{M}_{n}$s into account when allotting values to the $z_{n}$s—say, so that: $$\lim_{k\rightarrow\infty}\left|z_{n}\right|^{k}=\lim_{k\rightarrow\infty}\left|\det\mathbf{M}_{n}\right|^{k}$$ for all $n$ (assuming that none of the $\mathbf{M}_{n}s$ have a determinant of magnitude $1$)?

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