Here's the general set up, as I understand it. Recall that a character of $\mathbb{R}^{\times}$ is a continuous homomorphism from $\mathbb{R}^{\times}$ to $\mathbb{C}^{\times}$. Every character of $\mathbb{R}^{\times}$ is of the form $\operatorname{sgn}(x)^{\kappa} |x|^s$ for some $\kappa \in \{0,1\}$ and $s \in \mathbb{C}$.
Associated to a pair of characters $\operatorname{sgn}^{\kappa_1} |\cdot|^{s_1},\operatorname{sgn}^{\kappa_2} |\cdot|^{s_2}$ of $\mathbb{R}^{\times}$ is a principal series representation of $\mathrm{GL}_2(\mathbb{R})$. The vector space of this representation is called the induced model: it consists of smooth functions $f : \mathrm{GL}_2(\mathbb{R}) \to \mathbb{C}$ satisfying
$$f\left(\begin{pmatrix} a & b \\ 0 & d \end{pmatrix} g\right) = \operatorname{sgn}(a)^{\kappa_1} |a|^{s_1} \operatorname{sgn}(d)^{\kappa_2} |d|^{s_2} \left|\frac{a}{d}\right|^{1/2} f(g)$$
for all $a,d \in \mathbb{R}^{\times}$, $b \in \mathbb{R}$, and $g \in \mathrm{GL}_2(\mathbb{R})$. The group $\mathrm{GL}_2(\mathbb{R})$ acts on such functions via right-translation: $(h \cdot f)(g) = f(gh)$. In this way, we obtain a representation of $\mathrm{GL}_2(\mathbb{R})$, namely the principal series representation associated to this pair of characters. (It is the representation obtained by normalised parabolic induction from these characters.)
Such a principal series representation is usually, but not always, irreducible. Reducibility occurs only in two situations. The first is if $\operatorname{sgn}(x)^{\kappa_1} |x|^{s_1} \operatorname{sgn}(x)^{\kappa_2} |x|^{-s_2} = \operatorname{sgn}(x)^k |x|^{k - 1}$ for some positive integer $k \geq 2$ (i.e. $\kappa_1 + \kappa_2 \equiv k \pmod{2}$ and $s_1 - s_2 = k - 1$): there is an infinite-dimensional irreducible subrepresentation and a finite-dimensional irreducible subquotient. The second is if $\operatorname{sgn}(x)^{\kappa_1} |x|^{s_1} \operatorname{sgn}(x)^{\kappa_2} |x|^{-s_2} = \operatorname{sgn}(x)^k |x|^{1 - k}$ for some $k \geq 2$: there is an infinite-dimensional subquotient and finite-dimensional subrepresentation. In both cases, the irreducible subrepresentation or subquotient is a discrete series representation of weight $k$ twisted by a power of $\left|\det\right|$. (The twist is by $\left|\det\right|^{s_1 - \frac{k - 1}{2}} = \left|\det\right|^{s_2 + \frac{k - 1}{2}}$ for the former case and by $\left|\det\right|^{s_1 + \frac{k - 1}{2}} = \left|\det\right|^{s_2 - \frac{k - 1}{2}}$ for the latter case.)
Given a function $f$ in the induced model of such a principal series representation, let $F : \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{C}$ be the smooth function given by
$$F(x,y) = (-1)^{\kappa_1 + \kappa_2} \left|x^2 + y^2\right|^{-\frac{s_1 - s_2 + 1}{2}} f\begin{pmatrix} \frac{x}{\sqrt{x^2 + y^2}} & \frac{y}{\sqrt{x^2 + y^2}} \\ -\frac{y}{\sqrt{x^2 + y^2}} & \frac{x}{\sqrt{x^2 + y^2}}\end{pmatrix},$$
so that if $y \neq 0$,
$$F(x,y) = f\left(\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} y & -x \\ 0 & \frac{1}{y} \end{pmatrix}\right),$$
while
$$F(x,0) = f\begin{pmatrix} \frac{1}{x} & 0 \\ 0 & x \end{pmatrix}.$$
This function satisfies
$$F(ax,ay) = \operatorname{sgn}(a)^{\kappa_1 + \kappa_2} |a|^{s_2 - s_1 - 1} F(x,y)$$
for $a \in \mathbb{R}^{\times}$. In particular, it has parity $\varepsilon = (-1)^{\kappa_1 + \kappa_2}$ and is $w$-homogeneous for $w = s_2 - s_1 - 1$.
Conversely, any smooth function $F : \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{C}$ satisfying this property gives rise to a smooth function $f : \mathrm{GL}_2(\mathbb{R}) \to \mathbb{C}$ in the induced model: by the Iwasawa decomposition, every $g \in \mathrm{GL}_2(\mathbb{R})$ is of the form
$$g = \begin{pmatrix} a & b \\ 0 & d \end{pmatrix} \begin{pmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{pmatrix}$$
for some $a,d \in \mathbb{R}$, $b,\theta \in \mathbb{R}$, in which case we define
$$f(g) = \operatorname{sgn}(a)^{\kappa_1} |a|^{s_1} \operatorname{sgn}(d)^{\kappa_2} |d|^{s_2} \left|\frac{a}{d}\right|^{1/2} (-1)^{\kappa_1 + \kappa_2} F(\cos \theta,\sin \theta).$$
We call the vector space of such smooth functions $F$ the plane model of this principal series representation.
It remains to define an appropriate action of $\mathrm{GL}_2(\mathbb{R})$ on smooth functions $F : \mathbb{R}^2 \setminus \{(0,0)\} \to \mathbb{C}$ satisfying
$$F(ax,ay) = \operatorname{sgn}(a)^{\kappa_1 + \kappa_2} |a|^{s_2 - s_1 - 1} F(x,y)$$
that intertwines between these two models. This action is given by
$$(h \cdot F)(x,y) = \operatorname{sgn}(\det h)^{\kappa_2} \left|\det h\right|^{s_2 - \frac{1}{2}} F((x,y) \, {}^t h^{-1}).$$
You are interested in two special cases of the above, where the action is $(h \cdot F)(x,y) = F((x,y) \, {}^t h^{-1})$.
The first is the case $\kappa_1 = 0$, $\kappa_2 = 0$, $s_1 = -w - 1/2$, and $s_2 = 1/2$, so that the action of $\mathrm{GL}_2(\mathbb{R})$ on elements $F$ of the plane model is $(h \cdot F)(x,y) = F((x,y) \, {}^t h^{-1})$, while elements of the plane model are even and $w$-homogeneous, so that they satisfy $F(ax,ay) = |a|^w F(x,y)$ (i.e. the plane model is simply the vector space $V_w^1$). The associated principal series representation is irreducible so long as $-w - 1 \neq \pm (k - 1)$ for some $k \geq 2$ with $k \equiv 0 \pmod{2}$ (i.e. $w$ is not an even integer).
The second is $\kappa_1 = 1$, $\kappa_2 = 0$, $s_1 = -w - 1/2$, and $s_2 = 1/2$. The action of $\mathrm{GL}_2(\mathbb{R})$ on elements $F$ of the plane model is again $(h \cdot F)(x,y) = F((x,y) \, {}^t h^{-1})$, while elements of the plane model are instead odd and $w$-homogeneous, so that they satisfy $F(ax,ay) = \operatorname{sgn}(a) |a|^w F(x,y)$ (i.e. the plane model is $V_w^{-1}$). The associated principal series representation is irreducible so long as $-w - 1 \neq \pm (k - 1)$ for some $k \geq 2$ with $k \equiv 1 \pmod{2}$ (i.e. $w$ is not an odd integer except possibly $w = -1$).
