  We prove that the multiplicity of each irreducible component in the
$\mathcal{U}(\mathfrak{gl}_n)$-cyclic module generated by the $l$-th power
$\det^{(\alpha)}(X)^l$ of the $\alpha$-determinant is given by the rank of a
matrix whose entries are given by a variation of the spherical Fourier
transformation for $(\mathfrak{S}_{nl},\mathfrak{S}_l^n)$. Further, we
calculate the matrix explicitly when $n=2$. This gives not only another proof
of the result by Kimoto-Matsumoto-Wakayama (2007) but also a new aspect of the
representation theory of the $\alpha$-determinants.
