$O(2)^-$ is generated by the $SO(2)$ of rotations about the $z$-axis and a reflection through a vertical plane.
The space $V_l$ is generated by spherial harmonics, i.e., Cartan decomposition $$V_l=span\{Y_{l,m}\}_{m=-l}^{l}=H_0\oplus H_1 \oplus \cdots\oplus H_l\;(dim H_0=1,\;dim H_k=2 (k\geq 1))$$ where $ Y_{l,m}(\theta,\phi) = (-1)^m\sqrt{\frac{(2l+1)(l-m)!}{2(l+m)!}} \;P^m_l(\cos\theta) \frac{e^{im\phi}}{\sqrt{2\pi}} (m=-l,\cdots,l) $, the associated Legendre polynomial $P^m_l(z) = \frac1{2^ll!} (1-z^2)^{m/2} \frac{d^{l+m}}{dz^{l+m}}(z^2-1)^l $, and $\theta$ and $\phi$ represent colatitude and longitude, respectively. In particular, the colatitude $\theta$, or polar angle, ranges from $0$ at the North Pole, to $\pi/2$ at the Equator, to $\pi$ at the South Pole, and the longitude $\phi$, or azimuth, may assume all values with $0\leq\theta<2\pi$.
The monograph of Golubitsky-Stewart-Schaeffer(1988)[Vol.II, pp.126] says that the dimension of fixed-point subspace $Fix_{V_l}(O(2)^-)$ is $0$ as $l$ even and $1$ as $l$ odd, where $$Fix_{V_l}(O(2)^-)=\{v\in V_l| v(A^Tx)=v(x) \text{ for all } A\in O(2)^- \}.$$
Problem: I think that $Fix_{V_l}(SO(2))=\{ Y_{l,0}\}$, $ Y_{l,0}(\theta) = \sqrt{\frac{(2l+1)l!}{2l!}} \;P^0_l(\cos\theta) \frac{1}{\sqrt{2\pi}} $. Also, $ Y_{l,0}$ is independent of $\phi$ and is invariant under a reflection through a vertical plane. Hence I think that the dimension of $Fix_{V_l}(O(2)^-)$ is also $1$ as $l$ even.
So I want to know why the dimension of $Fix_{V_l}(O(2)^-)$ is $0$ as $l$ even. Where is wrong in my above analysis?