Some spherical function values for two-row tableaux and Young subgroups with three factors

A Young subgroup of the symmetric group $\mathcal{S}_{N}$ with three factors, is realized as the stabilizer $G_{n}$ of a monomial $x^{\lambda}$ ( $=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots x_{N}^{\lambda_{N}}$) with $\lambda=\left( d_{1}^{n_{1}},d_{2}^{n_{2}},d_{3}^{n_{3}}\right)$ (meaning $d_{j}$ is repeated $n_{j}$ times, $1\leq j\leq3$), thus is isomorphic to the direct product $\mathcal{S}_{n_{1}}\times\mathcal{S}_{n_{2}}\times \mathcal{S}_{n_{3}}$. The orbit of $x^{\lambda}$ under the action of $\mathcal{S}_{N}$ (by permutation of coordinates) spans a module $V_{\lambda}$, the representation induced from the identity representation of $G_{n}$. The space $V_{\lambda}$ decomposes into a direct sum of irreducible $\mathcal{S}$-modules. The spherical function is defined for each of these, it is the character of the module averaged over the group $G_{n}$. This paper concerns the value of certain spherical functions evaluated at a cycle which has no more than one entry in each of the three intervals $I_{j}=\left\{ i:\lambda_{i}=d_{j}\right\} ,1\leq j\leq3$. These values appear in the study of eigenvalues of the Heckman-Polychronakos operators in the paper by V. Gorin and the author (arXiv:2412:01938v1). The present paper determines the spherical function values for $\mathcal{S}_{N}$-modules $V$ of two-row tableau type, corresponding to Young tableaux of shape $\left[ N-k,k\right]$. The method is based on analyzing the effect of a cycle on $G_{n}$-invariant elements of $V$. These are constructed in terms of Hahn polynomials in two variables.