Representation theory of the $\alpha$-determinant and zonal spherical functions

Abstract

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.

9 pages