The hyperoctahedral group $H_n$ can be seen as the centralizer of the permutation $(12)(34)\cdots (2n-1\,2n)$ in $S_{2n}$. It has $2^nn!$ elements.

The quantities $$ \omega_\lambda(\pi)=\frac{1}{2^nn!}\sum_{h\in H_n}\chi_{2\lambda}(h\pi)$$ are called the zonal spherical functions of the Gelfand pair $(S_{2n},H_n)$. Here $2\lambda=(2\lambda_1,2\lambda_2,...)$ and $\chi$ are irreducible characters of $S$.

I have observed the following very nice result: $$ \frac{2^nn!}{(2n)!}\sum_{\lambda\vdash n}\chi_{2\lambda}(1^{2n})\omega_\lambda(\pi)=\begin{cases} 1, &\pi\in H_n\\0, &\pi\notin H_n\end{cases}$$

Does anyone know how to prove this? (Orthogonality of characters is not straightforwardly useful) If so, can it be generalized to $\sum_{\lambda\vdash n}\chi_{2\lambda}(\mu)\omega_\lambda(\pi)$?