Currently, I'm reading Kohnen's 'Newforms of half-integral weight'. In the proof of proposition 1 at page 7, Kohnen wrote
'' Thus, if $f \in S_{k+1 / 2}(N, \chi)$, we find $$ f|Q=\frac{1}{2} f|(Q+\tilde{Q})=\frac{1}{2}(f|\xi| \mathrm{Tr}+f|\xi^{\prime}| \mathrm{Tr})=\frac{\alpha}{4} f \mid \mathrm{Tr}=\alpha f $$ Conversely, suppose that $f \mid Q=\alpha f$. Then $$ f|(\xi-\frac{\alpha}{4})| \operatorname{Tr}=f|(\xi^{\prime}-\frac{\alpha}{4})| \operatorname{Tr}=0 $$ whence also $$ f|(\xi+\xi^{\prime}-\frac{\alpha}{2})| \operatorname{Tr}=0 $$ By definition of Tr the above equation means that the function $\hat{f}=f \mid(\xi+\xi^{\prime}-\frac{\alpha}{2})$ lies in the orthogonal complement of $S_{k+1 / 2}(4 N, \chi_{1})$ in $S_{k+1 / 2}(16 N, \chi_{1})$. ''
Why $f|Q=\frac{1}{2} f|(Q+\tilde{Q})$, is the action of $\xi$ and $\xi^{\prime}$ the same here? I also don't know how to prove ''the function $\hat{f}=f \mid(\xi+\xi^{\prime}-\frac{\alpha}{2})$ lies in the orthogonal complement of $S_{k+1 / 2}(4 N, \chi_{1})$ in $S_{k+1 / 2}(16 N, \chi_{1})$''. If there are some suggestions and reference about them? Thank you very much!