# Harmonic Analysis on a Finitely-Generated Matrix Group

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 $$1$$)?