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 $.