Let me preface this by saying that I have next to no background in representation theory. I come from geometry but the following representations showed up naturally in my work.


We let $ V = C^\infty \left( \mathbb{R}^2 \setminus \left\{ 0 \right\} \right) $ be equipped with its standard Frechet space topology. Then, $ V $ is a continuous representation of $ GL \left( 2, \mathbb{R} \right) $ via the action $$ gf \left( x \right) = f \left( g^{-1} x \right). $$ One easily sees that for any complex number $ w \in \mathbb{C} $ and any parity $ \varepsilon \in \left\{+,-\right\} $, the space $ V^\varepsilon\_w \subset V $ of smooth, $ w $-homogeneous and even (if $ \varepsilon = + $) resp. odd (if $ \varepsilon = - $) functions forms a closed subrepresentation of $ V $. My question now is: Is the representation $ V\_0^- $ irreducible? If someone knows the answer for general values of $ w $ and $ \varepsilon $, this would of course be even better.


My guess is that this question should somehow be connected to the so called principal series representations as these are also indexed by a complex number and a parity. I have looked this up in Knapp's book "Representation Theory of Semisimple Groups". However, the representations there look very different (at least to the untrained eye?!). Moreover, he only considers them as representations over $ SL(2,\mathbb{R}) $ while I am interested in the bigger group $ GL(2, \mathbb{R}) $.


EDIT: Instead of homogeneous of degree $ w $ I should have more precisely said positively homogeneous of degree $ w $, that is $ f (ax) = a^w f(x) $ for all $ a>0 $ and $ x \in \mathbb{R}^2 \setminus \left\{ 0 \right\} $. By even ($ \varepsilon = + $) resp. odd ($ \varepsilon = - $), I mean that $ f (-x) = \varepsilon f(x) $ for all $ x $.



Is the GL(2,R)-representation of smooth, odd and 0-homogeneous functions on the punctured plane irreducible?
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.)




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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.)




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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.




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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
$$(h \cdot F)(x,y) = \operatorname{sgn}(\det h)^{\kappa\_2} \left|\det h\right|^{s\_2 - \frac{1}{2}} F((x,y) \, {}^t h^{-1}).$$




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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 $w = -1$).


