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 $\sum\_{\lambda\vdash n}\chi\_{2\lambda}(\mu)\omega\_\lambda(\pi)$?



A sum over characters of $S_{2n}$ and zonal spherical functions of $(S_{2n},H_n)$
The essential thing here is that the characters $\chi\_{2\lambda}$ are exactly the irreducible constituents of the induced character $(1\_{H\_n})^{S\_n}$. The result generalizes to an arbitrary subgroup $H\leq G$ of a finite group $G$ as follows: For $x$, $y\in G$, we have $\DeclareMathOperator{\Irr}{Irr}$
$$ \frac{ |H| }{ |G| } 
 \sum\_{ \chi\in \Irr(G \mid 1\_H) }
 \chi(x^{-1} ) \omega\_{\chi}(y) 
 = \frac{ |x^G \cap Hy| }{ |x^G| }. \tag{\*}
$$
Here $\Irr(G \mid 1\_H)$ denotes the set of irreducible constituents of $(1\_H)^G$, which by Frobenius reciprocity is the set of $\chi\in \Irr(G)$ such that $1\_H$ is a constituent of the restriction $\chi\_H$.
As above, 
$$ \omega\_{\chi}(y) = \frac{1}{|H|}
 \sum\_{h\in H} \chi(hy),
$$ 
but we do not have to assume that $(G,H)$ is a Gelfand pair. ($x=1$ and $y=\pi$ is your result.) 


Proof: Let  

$$ e\_H = \frac{1}{|H|} \sum\_{h\in H} h \in \mathbb{C} H, 
$$
the central primitive idempotent of the group algebra belonging to $1\_H$. Then $\omega\_{\chi}(g) = \chi(e\_H g)$ by defintion. 
If $1\_H$ is not a constituent of the restriction $\chi\_H$ for $\chi\in \Irr(G)$, then $\chi(e\_Hg) = 0$ for all $g\in G$. 
Thus we can let run $\chi$ over all of $\Irr(G)$ in the sum in (\*). 
We get
$$ \frac{ |H| }{ |G| } 
 \sum\_{ \chi\in \Irr G } \chi( x^{-1} ) \omega\_{\chi}(y)
 = \frac{ 1 }{ |G| } 
 \sum\_{ h\in H } \sum\_{ \chi\in \Irr G } 
 \chi(x^{-1})\chi(hy), 
$$
and the second orthogonality relation for characters yields the result.


