Eigenvalues of supersymmetric Shimura operators and interpolation polynomials

The Shimura operators are a certain distinguished basis for invariant differential operators on a Hermitian symmetric space. Answering a question of Shimura, Sahi and Zhang showed that the Harish-Chandra images of these operators are specializations of certain $BC$-symmetric interpolation polynomials that were defined by Okounkov. We consider the analogs of Shimura operators for the Hermitian symmetric superpair $(\mathfrak{g},\mathfrak{k})$ where $\mathfrak{g}=\mathfrak{g}\mathfrak{l}(2p\mathrm{\mid }2q)$ and $\mathfrak{k}=\mathfrak{g}\mathfrak{l}(p\mathrm{\mid }q)\oplus \mathfrak{g}\mathfrak{l}(p\mathrm{\mid }q)$ and we prove their Harish-Chandra images are specializations of certain $BC$-supersymmetric interpolation polynomials introduced by Sergeev–Veselov.