Eigenvalues of supersymmetric Shimura operators and interpolation polynomials

Abstract

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 $(g,k)$ where $g=\mathrm{gl}\left(2p\right|2q)$ and $k=\mathrm{gl}\left(p\right|q)\oplus \mathrm{gl}(p\left|q\right)$ and we prove their Harish-Chandra images are specializations of certain BC-supersymmetric interpolation polynomials introduced by Sergeev and Veselov.