In my previous question, I asked about harmonic analysis on the group $\operatorname{SL}(3, \mathbb{R})$. The representation theory of this group appears to be quite complicated, so I am now looking at another group that is of interest in my application: the affine group $\operatorname{Aff}(2, \mathbb{R})$ consisting of matrices \begin{equation} A = (g, t) = \begin{bmatrix} g_{11} & g_{12} & t_1 \\ g_{21} & g_{22} & t_2 \\ 0 & 0 & 1 \end{bmatrix} \end{equation} where $g \in \operatorname{GL}(2, \mathbb{R})$ and $t \in \mathbb{R}^2$. If it simplifies things, I would also be interested in the restriction to $\operatorname{GL}^+(2, \mathbb{R})$ of $2 \times 2$ real matrices with strictly positive determinant (unlike $\operatorname{GL}(2,\mathbb{R})$, this group has only 1 connected component).
I would like to build an algorithm that performs the Fourier transform on this group, which is defined as $$ \hat{f}(\lambda) = \int_G f(g) U^\lambda(g^{-1}) d\mu(g) $$ where $U^{\lambda}$ is an irreducible unitary representation of $\operatorname{Aff}(2, \mathbb{R})$, and $\mu$ is a left-invariant Haar measure on this group (which is not unimodular).
Since we can only work with finitely sampled functions $f$, the way to go will be to assume that $f$ is "band limited", i.e. it is a linear combination of a finite number of matrix elements of irreducible representations. By increasing the number of matrix coefficients (the "resolution" of the algorithm), we can transform an increasingly rich class of functions.
Questions
As in my previous question, I'd like to know:
- Which IURs do I need in order to decompose a function in $L^2(G)$? I'm not set on this particular function space, so if another one is easier feel free to modify the question. I think this question amounts to finding a Plancherel theorem for this group.
- Are there any explicit formulas known for the matrix elements of the relevant IURs? What about the basis functions of irreducible representation spaces in $L^2(G)$? In both cases, integral representations are fine.
- Is anything known about the asymptotics of these functions? This is important because the IURs of $\operatorname{Aff}(2, \mathbb{R})$ are infinite-dimensional matrices, so an algorithm will have to choose some cutoff. If functions $U^\lambda_{mn}$ do not decay sufficiently fast (or at all) with $m,n$, that may be a showstopper.
Strategy
In response to my previous question, the user named guest mentioned that to deal with semidirect product groups (such as $\operatorname{Aff}(2,\mathbb{R}) = \mathbb{R}^2 \rtimes \operatorname{GL}(2, \mathbb{R})$), I should look at the concept of a system of imprimitivity. I have read a little bit on this topic, but do not yet understand how to apply it exactly. Intuitively I would think that Fourier analysis on the affine group should somehow "decompose" into Fourier analysis on its two factors, but I don't know if that is true or how to make this precise. Some guidance on how the system of imprimitivity concept helps to understand the representation theory of a semidirect product group would be very useful.
Provided that we can indeed understand the representation theory of $\operatorname{Aff}(2, \mathbb{R})$ by understanding the representation theory of $\mathbb{R}^2$ and that of $\operatorname{GL}(2, \mathbb{R})$ and then assembling them, the main problem is then to understand $\operatorname{GL}(2, \mathbb{R})$. A lot is known concretely about the representation theory of $\operatorname{SL}(2, \mathbb{R})$, but I could not find much on $\operatorname{GL}(2, \mathbb{R})$. I think that $\operatorname{GL}(2, \mathbb{R}) = \operatorname{SL}(2, \mathbb{R}) \times \mathbb{R}^*$, so maybe there is an easy way to extend the representation theory of $\operatorname{SL}(2, \mathbb{R})$ to $\operatorname{GL}(2, \mathbb{R})$?
I did find a paper by D. A. Vogan, "The unitary dual of GL(n) over an archimedean field.", but (without having read the paper) it seems a bit abstract and does not treat the Plancherel formula.
Thanks in advance!